OFFSET
0,4
COMMENTS
The subsequence of primes begins: 2, 7, 29, 127, 176557, 2177573, 151966597.
REFERENCES
A. Cayley, On the analytical forms called trees, with application to the theory of chemical combinations, Reports British Assoc. Advance. Sci. 45 (1875), 257-305 = Math. Papers, Vol. 9, 427-460 (see p. 450).
R. C. Read, personal communication.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
LINKS
T. D. Noe, Table of n, a(n) for n = 0..200
S. J. Cyvin, J. Brunvoll, and B. N. Cyvin, Enumeration of constitutional isomers of polyenes, J. Molec. Struct. (Theochem) 357, no. 3 (1995) 255-261.
R. C. Read, Letter to N. J. A. Sloane, Oct. 29, 1976
FORMULA
G.f.: A(x) = x*(1/3!)*(f^3+3*subs(x=x^2, f)*f+2*subs(x=x^3, f)), where f = G001190(x)/x, G001190 = g.f. for A001190.
Then g.f.: x*(cycle_index(S3, B0)+cycle_index(S3, G036656)+cycle_index(S3, G036657)+cycle_index(S2, B0)*(G036656+G036657)+cycle_index(S2, G036656)*(G036657+B0)+cycle_index(S2, G036657)*(B0+G036656)+B0*G036656*G036657), where cycle_index(Sk, f) means apply the cycle index for the symmetric group S_k to f(x).
E.g., cycle_index(S2, f) = (1/2!)*(f^2+subs(x=x^2, f), cycle_index(S3, f) = (1/3!)*(f^3+3*subs(x=x^2, f)*f+2*subs(x=x^3, f)).
MAPLE
N := 40: t1 := G001190/x: G000671 := series(x*(1/3!)*(t1^3+3*subs(x=x^2, t1)*t1+2*subs(x=x^3, t1)), x, N); A000671 := n->coeff(G000671, x, n);
CI2 := proc(f) (1/2)*(f^2+subs(x=x^2, f)); end; CI3 := proc(f) (1/6)*(f^3+3*subs(x=x^2, f)*f+2*subs(x=x^3, f)); end;
N := 40: B0 := series(1 + x, x, N): G000671 := series(x*(CI3(B0) + CI3(G036656) + CI3(G036657) + CI2(B0)*(G036656 + G036657) + CI2(G036656)*(G036657 + B0) + CI2(G036657)*(B0 + G036656) + B0*G036656*G036657), x, N); A036658 := n->coeff(G036658, x, n);
MATHEMATICA
terms = 32; (* B = g.f. for A001190 *) B[_] = 0; Do[B[x_] = x + (1/2)*(B[x]^2 + B[x^2]) + O[x]^terms // Normal, terms];
f[x_] = B[x]/x;
A[x_] = x*(1/3!)*(f[x]^3 + 3*f[x^2]*f[x] + 2*f[x^3]) + O[x]^terms;
CoefficientList[A[x], x] (* Jean-François Alcover, May 29 2012, from first g.f., updated Jan 10 2018 *)
CROSSREFS
KEYWORD
nonn,easy,nice
AUTHOR
STATUS
approved