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A028522
Duplicate of A007246.
+20
0
1, 0, 276, 2048, 11202, 49152, 184024
(list; graph; refs; listen; history; text; internal format)
OFFSET
-1,3
LINKS
Table of n, a(n) for n=-1..5.
KEYWORD
dead
STATUS
approved
A007191
McKay-Thompson series of class 2B for the Monster group with a(0) = -24.
(Formerly M5157)
+10
18
1, -24, 276, -2048, 11202, -49152, 184024, -614400, 1881471, -5373952, 14478180, -37122048, 91231550, -216072192, 495248952, -1102430208, 2390434947, -5061476352, 10487167336, -21301241856, 42481784514, -83300614144
(list; graph; refs; listen; history; text; internal format)
OFFSET
-1,2
COMMENTS
Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
Let t(q) = (eta(q) / eta(q^2))^24 = 1/q - 24 + 276q - 2048q^2 + ... If j(q) is the q-series for the j-invariant, with coefficients from A000521, then j(q) = (t + 256)^3/t^2 j(q^2) = (t + 16)^3/t. Hence t can be used to parametrize the classical modular curve X0(2). - Gene Ward Smith, Aug 04 2006
From Gary W. Adamson, Jun 06 2009: (Start)
Equals (1/q) * the convolution square of A161195: (1, -12, 66, -232, 639, ...)
and row sums of triangle A161196. (End)
Given g.f. A(q), Greenhill (1895) denotes -1/64 * A(q) by tau_oo on page 409 equation (43). - Michael Somos, Jul 17 2013
REFERENCES
J. H. Conway and S. P. Norton, Monstrous Moonshine, Bull. Lond. Math. Soc. 11 (1979) 308-339.
R. Fricke, Die elliptischen Funktionen und ihre Anwendungen, Teubner, 1922, Vol. 2, see p. 371. Eq. (1)
A. G. Greenhill, The Transformation and Division of Elliptic Functions, Proceedings of the London Mathematical Society (1895) 403-486.
G. Hoehn, Selbstduale Vertexoperatorsuperalgebren und das Babymonster, Bonner Mathematische Schriften, Vol. 286 (1996), 1-85.
J. McKay and H. Strauss, The q-series of monstrous moonshine and the decomposition of the head characters. Comm. Algebra 18 (1990), no. 1, 253-278.
S. Ramanujan, Modular Equations and Approximations to pi, pp. 23-39 of Collected Papers of Srinivasa Ramanujan, Ed. G. H. Hardy et al., AMS Chelsea 2000. See page 26.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
LINKS
Seiichi Manyama, Table of n, a(n) for n = -1..5000 (first 1001 terms from T. D. Noe)
R. E. Borcherds, Introduction to the monster Lie algebra, pp. 99-107 of M. Liebeck and J. Saxl, editors, Groups, Combinatorics and Geometry (Durham, 1990). London Math. Soc. Lect. Notes 165, Cambridge Univ. Press, 1992.
B. Brent, Quadratic Minima and Modular Forms, Experimental Mathematics, v.7 no.3, 257-274.
D. Ford, J. McKay and S. P. Norton, More on replicable functions, Commun. Algebra 22, No. 13, 5175-5193 (1994).
G. Hoehn (gerald(AT)math.ksu.edu), Selbstduale Vertexoperatorsuperalgebren und das Babymonster, Doctoral Dissertation, Univ. Bonn, Jul 15 1995 (pdf, ps).
Michael Somos, Emails to N. J. A. Sloane, 1993
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions
Index entries for sequences related to groups
Index entries for McKay-Thompson series for Monster simple group
FORMULA
G.f.: (1/x)(Product_{k>0} 1/(1 + x^k))^24.
G.f.: (1/q)(Product_{k>0} (1 - q^(2*k - 1)))^24 = 64 * (g_n)^24 where q = e^(-Pi sqrt(n)) and g_n is Ramanujan's class invariant.
(eta(q)/eta(q^2))^24. - Gene Ward Smith, Aug 04 2006
Expansion of q^(-1) * chi(-q)^24 in powers of q where chi() is a Ramanujan theta function. - Michael Somos, Aug 19 2007
Euler transform of period 2 sequence [-24, 0, ...]. - Michael Somos, Aug 19 2007
Expansion of (1 - lambda(t)) / (lambda(t) / 16)^2 in powers of q = exp(2 Pi i t) where lambda() is the elliptic modular function A115977. - Michael Somos, Aug 19 2007
Expansion of 64 tau(omega) in powers of q = exp(2 Pi i omega) where tau() is Fricke's function on page 371 equation (1). - Michael Somos, Jun 12 2012
G.f. A(x) satisfies 0 = f(A(x), A(x^2)) where f(u, v) = u^2*v - v^2 + 48*u*v + 4096*u. - Michael Somos, Aug 19 2007
G.f. is a period 1 Fourier series which satisfies f(-1 / (2 t)) = 4096 g(t) where q = exp(2 Pi i t) and g() is the g.f. of A014103. - Michael Somos, Aug 19 2007
a(n) = -(-1)^n * A097340(n). A007246(n) = a(n) unless n = 0.
Convolution inverse of A014103.
a(n) ~ -(-1)^n * exp(2*Pi*sqrt(n)) / (2 * n^(3/4)). - Vaclav Kotesovec, Aug 27 2015
a(-1) = 1, a(n) = -(24/(n+1))*Sum_{k=1..n+1} A000593(k)*a(n-k) for n > -1. - Seiichi Manyama, Mar 29 2017
EXAMPLE
G.f. = 1/q - 24 + 276*q - 2048*q^2 + 11202*q^3 - 49152*q^4 + 184024*q^5 - ...
MATHEMATICA
a[ n_] := SeriesCoefficient[ QPochhammer[ q, q^2]^24 / q, {q, 0, n}]; (* Michael Somos, Jul 11 2011 *)
a[ n_] := SeriesCoefficient[ Product[ 1 - q^k, {k, 1, n + 1, 2}]^24 / q, {q, 0, n}]; (* Michael Somos, Jul 11 2011 *)
a[ n_] := With[ {m = ModularLambda[ Log[q] / (Pi I)]}, SeriesCoefficient[ (1 - m) / (m/16)^2, {q, 0, 2 n}]]; (* Michael Somos, Jul 11 2011 *)
a[ n_] := With[ {m = InverseEllipticNomeQ @ q}, SeriesCoefficient[ (1 - m) / (m/16)^2, {q, 0, 2 n}]]; (* Michael Somos, Jul 11 2011 *)
PROG
(PARI) {a(n) = if( n<-1, 0, n++; polcoeff( prod( k=1, n, 1 + x^k, 1 + x * O(x^n))^-24, n))};
(PARI) {a(n) = my(A); if( n<-1, 0, n++; A = x * O(x^n); polcoeff( (eta(x + A) / eta(x^2 + A))^24, n))};
CROSSREFS
A134786, A045479, A007191, A097340, A035099, A007246, A107080 are all essentially the same sequence.
Cf. A161195, A161196, A014103, A115977.
KEYWORD
sign,easy,nice
AUTHOR
N. J. A. Sloane
STATUS
approved
A097340
McKay-Thompson series of class 4A for the Monster group with a(0) = 24.
+10
14
1, 24, 276, 2048, 11202, 49152, 184024, 614400, 1881471, 5373952, 14478180, 37122048, 91231550, 216072192, 495248952, 1102430208, 2390434947, 5061476352, 10487167336, 21301241856, 42481784514, 83300614144
(list; graph; refs; listen; history; text; internal format)
OFFSET
-1,2
COMMENTS
Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
This series is also called Weber's modular function. - N. J. A. Sloane, Jun 23 2011
Or, better, it is the 24th power of Weber's modular function f(). - Michael Somos, Jan 10 2017
Given g.f. A(q), Greenhill (1895) denotes 1/64 * A(q) by tau_1 on page 409 equation (43). - Michael Somos, Jul 17 2013
REFERENCES
S. Ramanujan, Modular Equations and Approximations to Pi, pp. 23-39 of Collected Papers of Srinivasa Ramanujan, Ed. G. H. Hardy et al., AMS Chelsea 2000. See page 26.
LINKS
T. D. Noe, Table of n, a(n) for n = -1..1000
Barry Brent, Finite Field Models of Polynomials Interpolating Fourier Coefficients of Modular Functions for Hecke Groups, Integers (2024) Vol. 24, Art. No. A18.
A. G. Greenhill, The Transformation and Division of Elliptic Functions, Proceedings of the London Mathematical Society (1895) 403-486.
Vaclav Kotesovec, A method of finding the asymptotics of q-series based on the convolution of generating functions, arXiv:1509.08708 [math.CO], 2015-2016.
Titus Piezas III, Pi Formulas, Ramanujan, and the Baby Monster Group
Titus Piezas III, Ramanujan's Constant exp(Pi sqrt(163)) And Its Cousins
Titus Piezas III, 0011: Article 1 (The j-function) - A collection of Algebraic Identities
Titus Piezas III, 0022: The 163 Dimensions - A collection of Algebraic Identities
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions
Eric Weisstein's World of Mathematics, Elliptic Lambda Function
FORMULA
Expansion of q^(-1) * chi(q)^24 where chi() is a Ramanujan theta function.
Expansion of (eta(q^2)^2 / (eta(q) * eta(q^4)))^24 in powers of q.
Euler transform of period 4 sequence [ 24, -24, 24, 0, ...].
G.f. is Fourier series of a level 4 modular function. f(-1 / (4 t)) = f(t) where q = exp(2 Pi i t).
G.f. A(x) satisfies 0 = f(A(x), A(x^2)) where f(u, v) = u*v * (u^3 + v^3) + (-u^3 + 48*u^2 - 96*u) * v^3 + (48*u^3 + 1791*u^2 + 2352*u) * v^2 + (-96*u^3 + 2352*u^2 - 10496*u) * v + 4096.
G.f. (1/q) * (Product_{k>0} (1 + q^(2k-1)))^24 = 64 * (G_n)^24 where q = e^(-Pi sqrt(n)) and G_n is a Ramanujan class invariant.
A007191(n) = -(-1)^n * a(n).
a(n) ~ exp(2*Pi*sqrt(n)) / (2 * n^(3/4)). - Vaclav Kotesovec, Aug 27 2015
From Peter Bala, Sep 25 2023: (Start)
Laurent series g.f.: A(q) = - 16*lambda(-q)/lambda(q)^2 = 1/q + 24 + 276*q + 2048*q^2 + ..., where lambda(q) = 16*q - 128*q^2 + 704*q^3 - 3072*q^4 + ... is the elliptic modular function in powers of the nome q = exp(i*Pi*t), the g.f. of A115977; lambda(q) = k(q)^2, where k(q) = (theta_2(q) / theta_3(q))^2 is the elliptic modulus.
A(q) = -16*(1 - lambda(-q))^2/lambda(-q). (End)
EXAMPLE
G.f. = 1/q + 24 + 276*q + 2048*q^2 + 11202*q^3 + 49152*q^4 + 184024*q^5 + ...
MATHEMATICA
a[ n_] := With[ {m = InverseEllipticNomeQ @ q}, SeriesCoefficient[ ((1 - m) m / 16)^-1, {q, 0, n}]]; (* Michael Somos, Jul 11 2011 *)
a[ n_] := SeriesCoefficient[ Product[1 + q^k, {k, 1, n + 1, 2}]^24 / q, {q, 0, n}]; (* Michael Somos, Jul 11 2011 *)
a[ n_] := SeriesCoefficient[ QPochhammer[ -q, q^2]^24 / q, {q, 0, n}]; (* Michael Somos, Nov 04 2014 *)
nmax = 50; CoefficientList[Series[Product[(1 + x^(2*k+1))^24, {k, 0, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Aug 27 2015 *)
PROG
(PARI) {a(n) = my(A); if( n<-1, 0, n++; A = x^n * O(x); polcoeff( (eta(x^2 + A)^2 / (eta(x + A) * eta(x^4 + A)))^24, n))};
CROSSREFS
A007191, A007246, A045479, A035099, A097340, A107080, A134786 are all essentially the same sequence.
Cf. A000700, A115977.
KEYWORD
nonn
AUTHOR
Michael Somos, Aug 05 2004
STATUS
approved
A107080
McKay-Thompson series of class 4A for the Monster group.
+10
8
1, 0, 276, 2048, 11202, 49152, 184024, 614400, 1881471, 5373952, 14478180, 37122048, 91231550, 216072192, 495248952, 1102430208, 2390434947, 5061476352, 10487167336, 21301241856, 42481784514, 83300614144, 160791890304, 305854488576, 573872089212, 1063005978624, 1945403602764, 3519965179904
(list; graph; refs; listen; history; text; internal format)
OFFSET
-1,3
COMMENTS
Also character of extremal vertex operator algebra of rank 12.
LINKS
Seiichi Manyama, Table of n, a(n) for n = -1..10000 (terms -1..1000 from T. D. Noe)
D. Ford, J. McKay and S. P. Norton, More on replicable functions, Commun. Algebra 22, No. 13, 5175-5193 (1994).
G. Hoehn (gerald(AT)math.ksu.edu), Selbstduale Vertexoperatorsuperalgebren und das Babymonster, Doctoral Dissertation, Univ. Bonn, Jul 15 1995 (pdf, ps).
G. Hoehn, Selbstduale Vertexoperatorsuperalgebren und das Babymonster, arXiv:0706.0236 [math.QA], 2007, from Bonner Mathematische Schriften, Vol. 286 (1996), 1-85.
Index entries for McKay-Thompson series for Monster simple group
FORMULA
G.f.: (1/x)(Product_{k>0} (1+x^k)/(1+x^(2k)))^24 -24.
a(n) = -(-1)^n * A007246(n).
a(n) ~ exp(2*Pi*sqrt(n)) / (2*n^(3/4)). - Vaclav Kotesovec, Sep 06 2015
EXAMPLE
T4A = 1/q + 276q + 2048q^2 + 11202q^3 + 49152q^4 + 184024q^5 +...
MATHEMATICA
a[0] = 0; a[n_] := SeriesCoefficient[ Product[1 - q^k, {k, 1, n+1, 2}]^24/q, {q, 0, n}] // Abs; Table[a[n], {n, -1, 20}] (* Jean-François Alcover, Oct 14 2013, after Michael Somos *)
QP = QPochhammer; s = (QP[q^2]^2/QP[q]/QP[q^4])^24 - 24*q + O[q]^30; CoefficientList[s, q] (* Jean-François Alcover, Nov 15 2015, after Michael Somos *)
PROG
(PARI) {a(n) = my(A); if( n<-1, 0, n++; A = x * O(x^n); polcoeff( (eta(x^2 + A)^2 / (eta(x + A) * eta(x^4 + A)))^24 - 24*x, n))};
(PARI) q='q+O('q^66); Vec(+40*q+(eta(q)^4 / eta(q^4)^4 - q*4^2*eta(q^4)^4 / eta(q)^4)^2) \\ Joerg Arndt, Mar 23 2017
CROSSREFS
Cf. A007246.
A134786, A045479, A007191, A097340, A035099, A007246, A107080 are all essentially the same sequence.
KEYWORD
nonn
AUTHOR
Michael Somos, May 11 2005
STATUS
approved
A035099
McKay-Thompson series of class 2B for the Monster group with a(0) = 40.
+10
7
1, 40, 276, -2048, 11202, -49152, 184024, -614400, 1881471, -5373952, 14478180, -37122048, 91231550, -216072192, 495248952, -1102430208, 2390434947, -5061476352, 10487167336, -21301241856, 42481784514, -83300614144
(list; graph; refs; listen; history; text; internal format)
OFFSET
-1,2
COMMENTS
Also Fourier coefficients of j_2 where j_2 is an analytic isomorphism H/\Gamma_0(2) ->\hat{C}.
"The function j_2 is analogous to j because it is modular (weight zero) for \Gamma_0(2), holomorphic on the upper half-plane, has a simple pole at infinity, generates the field of \Gamma_0(2)-modular functions, and defines a bijection of a \Gamma_0(2) fundamental set with C." from the Brent article page 260 using his notation of j_2. - Michael Somos, Mar 08 2011
Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
REFERENCES
G. Hoehn, Selbstduale Vertexoperatorsuperalgebren und das Babymonster, Bonner Mathematische Schriften, Vol. 286 (1996), 1-85.
LINKS
T. D. Noe, Table of n, a(n) for n=-1..1000
R. E. Borcherds, Introduction to the monster Lie algebra, pp. 99-107 of M. Liebeck and J. Saxl, editors, Groups, Combinatorics and Geometry (Durham, 1990). London Math. Soc. Lect. Notes 165, Cambridge Univ. Press, 1992.
B. Brent, Quadratic Minima and Modular Forms, Experimental Mathematics, v.7 no.3, 257-274; see also Project Euclid
J. H. Conway and S. P. Norton, Monstrous Moonshine, Bull. Lond. Math. Soc. 11 (1979) 308-339.
D. Ford, J. McKay and S. P. Norton, More on replicable functions, Commun. Algebra 22, No. 13, 5175-5193 (1994).
G. Hoehn (gerald(AT)math.ksu.edu), Selbstduale Vertexoperatorsuperalgebren und das Babymonster, Doctoral Dissertation, Univ. Bonn, Jul 15 1995 (pdf, ps).
Masao Koike, Modular forms on non-compact arithmetic triangle groups, Unpublished manuscript [Extensively annotated with OEIS A-numbers by N. J. A. Sloane, Feb 14 2021. I wrote 2005 on the first page but the internal evidence suggests 1997.]
J. McKay and H. Strauss, The q-series of monstrous moonshine and the decomposition of the head characters, Comm. Algebra 18 (1990), no. 1, 253-278.
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Monster Group
Index entries for "core" sequences
FORMULA
Expansion of 64 + q^(-1) * (phi(-q) / psi(q))^8 in powers of q where phi(), psi() are Ramanujan theta functions. - Michael Somos, Mar 08 2011
Expansion of 64 + (eta(q) / eta(q^2))^24 in powers of q. - Michael Somos, Mar 08 2011
j_2 = E_{gamma, 2}^2 / E_{oo, 4} in the notation of Brent where E_{gamma, 2} is g.f. for A004011 and E_{oo, 4} is g.f. for A007331. - Michael Somos, Mar 08 2011
G.f.: 64 + x^(-1) * (Product_{k>0} 1 + x^k)^(-24). - Michael Somos, Mar 08 2011
a(n) ~ (-1)^(n+1) * exp(2*Pi*sqrt(n)) / (2*n^(3/4)). - Vaclav Kotesovec, Nov 16 2016
EXAMPLE
j_2 = 1/q + 40 + 276*q - 2048*q^2 + 11202*q^3 - 49152*q^4 + 184024*q^5 + ...
MATHEMATICA
max = 21; f[x_] := Product[ 1 + x^k, {k, 1, max}]^(-24); coes = CoefficientList[ Series[ f[x], {x, 0, max} ], x]; a[n_] := coes[[n+2]]; a[0] = 40; Table[a[n], {n, -1, max-1}] (* Jean-François Alcover, Nov 03 2011, after Michael Somos *)
QP = QPochhammer; s = 64*q + (QP[q]/QP[q^2])^24 + O[q]^30; CoefficientList[s, q] (* Jean-François Alcover, Nov 15 2015, after Michael Somos *)
PROG
(PARI) {a(n) = my(A); if( n<-1, 0, n++; A = x * O(x^n); polcoeff( 64 * x + (eta(x + A) / eta(x^2 + A))^24, n))}; /* Michael Somos, Mar 08 2011 */
CROSSREFS
Cf. A134786, A045479, A007191, A097340, A035099, A007246, A107080 are all essentially the same sequence.
KEYWORD
easy,sign,nice,core
AUTHOR
Barry Brent (barryb(AT)primenet.com)
STATUS
approved
A045479
McKay-Thompson series of class 2B for the Monster group with a(0) = -8.
+10
7
1, -8, 276, -2048, 11202, -49152, 184024, -614400, 1881471, -5373952, 14478180, -37122048, 91231550, -216072192, 495248952, -1102430208, 2390434947, -5061476352, 10487167336, -21301241856, 42481784514, -83300614144
(list; graph; refs; listen; history; text; internal format)
OFFSET
-1,2
COMMENTS
Unsigned sequence gives McKay-Thompson series of class 4A for Monster; also character of extremal vertex operator algebra of rank 12.
The value of a(0) is the Rademacher constant for the modular function and appears in Conway and Norton's Table 4. - Michael Somos, Mar 08 2011
REFERENCES
G. Hoehn, Selbstduale Vertexoperatorsuperalgebren und das Babymonster, Bonner Mathematische Schriften, Vol. 286 (1996), 1-85.
LINKS
T. D. Noe, Table of n, a(n) for n = -1..1000
R. E. Borcherds, Introduction to the monster Lie algebra, pp. 99-107 of M. Liebeck and J. Saxl, editors, Groups, Combinatorics and Geometry (Durham, 1990). London Math. Soc. Lect. Notes 165, Cambridge Univ. Press, 1992.
B. Brent, Quadratic Minima and Modular Forms, Experimental Mathematics, v.7 no.3, 257-274.
J. H. Conway and S. P. Norton, Monstrous Moonshine, Bull. Lond. Math. Soc. 11 (1979) 308-339.
D. Ford, J. McKay and S. P. Norton, More on replicable functions, Commun. Algebra 22, No. 13, 5175-5193 (1994).
G. Hoehn (gerald(AT)math.ksu.edu), Selbstduale Vertexoperatorsuperalgebren und das Babymonster, Doctoral Dissertation, Univ. Bonn, Jul 15 1995 (pdf, ps).
J. McKay and H. Strauss, The q-series of monstrous moonshine and the decomposition of the head characters, Comm. Algebra 18 (1990), no. 1, 253-278.
Index entries for McKay-Thompson series for Monster simple group
FORMULA
Expansion of 16 + (eta(q) / eta(q^2))^24 in powers of q. - Michael Somos, Mar 08 2011
a(n) ~ (-1)^(n+1) * exp(2*Pi*sqrt(n)) / (2*n^(3/4)). - Vaclav Kotesovec, Sep 07 2017
EXAMPLE
1/q - 8 + 276*q - 2048*q^2 + 11202*q^3 - 49152*q^4 + 184024*q^5 + ...
MATHEMATICA
a[0] = -8; a[n_] := SeriesCoefficient[ Product[1 - q^k, {k, 1, n+1, 2}]^24/q, {q, 0, n}]; Table[a[n], {n, -1, 20}] (* Jean-François Alcover, Oct 14 2013, after Michael Somos *)
QP = QPochhammer; s = 16*q + (QP[q]/QP[q^2])^24 + O[q]^30; CoefficientList[s, q] (* Jean-François Alcover, Nov 15 2015, after Michael Somos *)
PROG
(PARI) {a(n) = my(A); if( n<-1, 0, n++; A = x * O(x^n); polcoeff( 16 * x + (eta(x + A) / eta(x^2 + A))^24, n))}; /* Michael Somos, Mar 08 2011 */
CROSSREFS
A134786, A045479, A007191, A097340, A035099, A007246, A107080 are all essentially the same sequence.
KEYWORD
sign,easy,nice
AUTHOR
N. J. A. Sloane
STATUS
approved
A134786
McKay-Thompson series of class 4A for the Monster group with a(0) = 4.
+10
7
1, 4, 276, 2048, 11202, 49152, 184024, 614400, 1881471, 5373952, 14478180, 37122048, 91231550, 216072192, 495248952, 1102430208, 2390434947, 5061476352, 10487167336, 21301241856, 42481784514, 83300614144, 160791890304
(list; graph; refs; listen; history; text; internal format)
OFFSET
-1,2
LINKS
G. C. Greubel, Table of n, a(n) for n = -1..1000
M. Koike, Mathieu group M24 and modular forms, Nagoya Math. J., 99 (1985), 147-157. MR0805086 (87e:11060)
Index entries for McKay-Thompson series for Monster simple group
FORMULA
Associated with permutations in Mathieu group M24 of shape (4)^4(2)^2(1)^4.
G.f. is a period 1 Fourier series which satisfies f(-1 / (4 t)) = f(t) where q = exp(2 Pi i t).
a(n) = A107080(n) unless n=0. Convolution with A030212 is A037219.
a(n) ~ exp(2*Pi*sqrt(n)) / (2*n^(3/4)). - Vaclav Kotesovec, Sep 07 2017
EXAMPLE
G.f. = 1/q + 4 + 276*q + 2048*q^2 + 11202*q^3 + 49152*q^4 + 184024*q^5 + ...
MATHEMATICA
a[0] = 4; a[n_] := SeriesCoefficient[ Product[1 - q^k, {k, 1, n+1, 2}]^24/q, {q, 0, n}] // Abs; Table[a[n], {n, -1, 21}] (* Jean-François Alcover, Oct 14 2013, after Michael Somos *)
QP = QPochhammer; s = (QP[q^2]^2/QP[q]/QP[q^4])^24 - 20*q + O[q]^30; CoefficientList[s, q] (* Jean-François Alcover, Nov 15 2015, after Michael Somos *)
a[ n_] := SeriesCoefficient[ -20 + QPochhammer[ -q, q^2]^24 / q, {q, 0, n}]; (* Michael Somos, May 05 2016 *)
PROG
(PARI) {a(n) = my(A); if( n<-1, 0, A = x^2 * O(x^n); A = (eta(x + A) / eta(x^4 + A))^8 / x; polcoeff( 12 + A + 256 / A, n))};
CROSSREFS
Cf. A030212, A037219, A107080.
Cf. A097340. [From R. J. Mathar, Dec 13 2008]
A134786, A045479, A007191, A097340, A035099, A007246, A107080 are all essentially the same sequence.
KEYWORD
nonn
AUTHOR
Michael Somos, Nov 22 2007
STATUS
approved
A028532
Character of extremal vertex operator algebra of rank 23/2.
+10
0
1, 0, 276, 1771, 9430, 39445, 142531, 460391, 1370156, 3810341, 10013717, 25082282, 60303447, 139869762, 314255118, 686285408, 1461010508, 3039221633, 6190257789, 12366731770, 24269856335, 46851441255, 89069526921, 166930973477, 308709141202, 563802228832
(list; graph; refs; listen; history; text; internal format)
OFFSET
0,3
REFERENCES
G. Hoehn, Selbstduale Vertexoperatorsuperalgebren und das Babymonster, Bonner Mathematische Schriften, Vol. 286 (1996), 1-85.
LINKS
Table of n, a(n) for n=0..25.
G. Hoehn (gerald(AT)math.ksu.edu), Selbstduale Vertexoperatorsuperalgebren und das Babymonster, Doctoral Dissertation, Univ. Bonn, Jul 15 1995 (pdf, ps).
FORMULA
G.f.: x^(2*r/24) * (B(x)^(2*r) - 2*r*B(x)^(2*r-24) where B(x) = x^(-1/24) * Product_{k>=0} (1+x^(2*k+1)) = x^(-1/24) * A000700 and r = 23/2. - Sean A. Irvine, Feb 29 2020
CROSSREFS
Cf. A000700, A007246.
KEYWORD
nonn,easy
AUTHOR
N. J. A. Sloane.
EXTENSIONS
More terms from Sean A. Irvine, Feb 29 2020
STATUS
approved
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