A306760 -id:A306760

A306769

Decimal expansion of Sum_{k>=2} (-1)^k * Zeta(k)^2 / k.

+10
5

1, 0, 4, 3, 4, 0, 2, 9, 1, 7, 5, 7, 4, 2, 8, 8, 7, 3, 3, 2, 5, 5, 2, 8, 9, 6, 4, 6, 6, 7, 1, 6, 7, 6, 0, 3, 0, 5, 4, 8, 4, 7, 0, 8, 6, 6, 0, 4, 6, 8, 8, 2, 5, 6, 1, 0, 4, 4, 5, 7, 0, 4, 7, 9, 7, 6, 9, 5, 8, 5, 0, 6, 2, 5, 5, 2, 5, 2, 4, 8, 4, 3, 2, 7, 6, 1, 5, 1, 0, 7, 2, 0, 7, 9, 8, 4, 1, 4, 3, 5, 6, 2, 1, 4, 6

(list; constant; graph; refs; listen; history; text; internal format)

OFFSET

1,3

COMMENTS

Sum_{k>=2} (-1)^k*Zeta(k)/k = A001620 (see MathWorld, formula 122).

LINKS

Table of n, a(n) for n=1..105.

Eric Weisstein's MathWorld, Riemann Zeta Function

Wikipedia, Riemann Zeta Function

FORMULA

Equals log(A306765) + A001620^2.

EXAMPLE

1.043402917574288733255289646671676030548470866046882561044570479769585...

MAPLE

evalf(Sum((-1)^j*Zeta(j)^2/j, j=2..infinity), 100);

MATHEMATICA

NSum[(-1)^k*Zeta[k]^2/k, {k, 2, Infinity}, WorkingPrecision -> 200, NSumTerms -> 100000]

PROG

(PARI) sumalt(k=2, (-1)^k*zeta(k)^2/k) \\ Michel Marcus, Mar 09 2019

CROSSREFS

Cf. A231132, A306760, A306765, A306774, A306778, A307106.

KEYWORD

nonn,cons

AUTHOR

Vaclav Kotesovec, Mar 09 2019

STATUS

approved

A306765

Decimal expansion of lim_{k->oo} (k^A001620 / k!) * Product_{j=1..k} Gamma(1/j).

+10
3

2, 0, 3, 4, 4, 4, 8, 9, 4, 5, 4, 8, 7, 6, 1, 6, 4, 7, 7, 9, 8, 0, 3, 5, 5, 5, 3, 1, 8, 8, 6, 9, 0, 2, 6, 3, 5, 5, 9, 7, 9, 4, 3, 9, 8, 6, 3, 7, 0, 2, 3, 7, 6, 2, 6, 0, 0, 0, 5, 2, 8, 4, 1, 6, 5, 6, 5, 0, 0, 7, 8, 2, 7, 7, 5, 7, 1, 1, 3, 2, 4, 4, 5, 0, 2, 6, 5, 0, 4, 0, 6, 1, 3, 5, 0, 7, 5, 0, 2, 9, 1, 2, 7, 1, 4

(list; constant; graph; refs; listen; history; text; internal format)

OFFSET

1,1

LINKS

Vaclav Kotesovec, Table of n, a(n) for n = 1..297

FORMULA

Equals exp(-gamma^2 + Sum_{j>=2} (-1)^j*Zeta(j)^2/j), where gamma is the Euler-Mascheroni constant A001620.

Equals exp(-gamma^2 + A306769).

Equals lim_{k->oo} k^(k*(2*k+1) + 2*gamma) * (2*Pi)^k * exp(1/6 + log(k)^2 - 2*k^2) / A306760(k).

EXAMPLE

2.0344489454876164779803555318869026355979439863702376260005284165650078277571...

MAPLE

evalf(exp(-gamma^2 + Sum((-1)^j*Zeta(j)^2/j, j=2..infinity)), 100);

MATHEMATICA

slogam = Table[Sum[LogGamma[1/j], {j, 1, n}], {n, 1, 1000}]; $MaxExtraPrecision = 1000; funs[n_] := E^slogam[[n]] * n^EulerGamma/n!; Do[Print[N[Sum[(-1)^(m + j) * funs[j*Floor[Length[slogam]/m]] * (j^(m - 1)/(j - 1)!/(m - j)!), {j, 1, m}], 80]], {m, 10, 100, 10}]

PROG

(PARI) exp(-Euler^2 + sumalt(j=2, (-1)^j*zeta(j)^2/j))

CROSSREFS

Cf. A001620, A231132, A303670, A306760, A306769.

KEYWORD

nonn,cons

AUTHOR

Vaclav Kotesovec, Mar 08 2019

STATUS

approved

A306789

a(n) = Product_{k=0..n} binomial(n + k, n).

+10
3

1, 2, 18, 800, 183750, 224042112, 1475939646720, 53195808994099200, 10587785727897772143750, 11721562427290210695200000000, 72596493516095364770534596279431168, 2527156530619699341247423878706695556300800, 496395279097923766533851314609410101501472675840000

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,2

COMMENTS

Sum_{k=0..n} binomial(n + k, n) = binomial(2*n + 1, n).

Product_{k=1..n} binomial(k*n, n) = (n^2)! / (n!)^n.

LINKS

Table of n, a(n) for n=0..12.

FORMULA

a(n) = (n+1)^n * BarnesG(2*n+2) / (Gamma(n+2)^n * BarnesG(n+2)^2).

a(n) ~ A * 2^(2*n^2 + 3*n/2 - 1/12) / (exp(n^2/2 + 1/6) * Pi^((n+1)/2) * n^(n/2 + 5/12)), where A is the Glaisher-Kinkelin constant A074962.

a(n) = a(n-1)*2n*(2n-1)!^2/(n!^3*n^(n-1)). - Chai Wah Wu, Jun 26 2023

MATHEMATICA

Table[Product[Binomial[n+k, n], {k, 0, n}], {n, 0, 13}]

Table[(n+1)^n * BarnesG[2*n+2] / (Gamma[n+2]^n * BarnesG[n+2]^2), {n, 0, 13}]

PROG

(Python)

from math import factorial

from functools import lru_cache

@lru_cache(maxsize=None)

def A306789(n): return A306789(n-1)*2*n*factorial(2*n-1)**2//factorial(n)**3//n**(n-1) if n else 1 # Chai Wah Wu, Jun 26 2023

CROSSREFS

Cf. A001700, A306760.

KEYWORD

nonn

AUTHOR

Vaclav Kotesovec, Mar 10 2019

STATUS

approved

A306907

a(n) = Product_{i=0..n, j=0..n, k=0..n} (i*j*k + 1).

+10
3

1, 2, 60750, 193002701276968128000000, 5076574217867350877310882935055477754989937924247841796875000000000000

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,2

COMMENTS

Next term is too long to be included.

LINKS

Table of n, a(n) for n=0..4.

FORMULA

a(n) = (n!)^(3*n^2) * Product_{i=1..n, j=1..n, k=1..n} (1 + 1/(i*j*k)).

a(n) ~ exp(3*n^2*log(Gamma(n+1)) + (gamma + PolyGamma(0, n+1))^3 - c), where gamma is the Euler-Mascheroni constant A001620 and c = A307106 = Sum_{k>=2} (-1)^k * Zeta(k)^3 / k = 1.836921908595663783265640880112170343162564662453544904457...

a(n) ~ (2*Pi)^(3*n^2/2) * exp(-3*n^3 + n/4 + (log(n))^3 + 3*gamma*(log(n))^2 + gamma^3 - c) * n^(3*(n^3 + n^2/2 + gamma^2)).

MATHEMATICA

Table[Product[i*j*k+1, {i, 0, n}, {j, 0, n}, {k, 0, n}], {n, 0, 5}]

CROSSREFS

Cf. A306760, A307106.

KEYWORD

nonn

AUTHOR

Vaclav Kotesovec, Mar 25 2019

STATUS

approved

A324589

a(n) = Product_{j=1..n, k=1..n} (1 + (j*k)^2).

+10
3

1, 2, 850, 9541930000, 62954953875193006250000, 2232026314050243695025069057306526600000000, 2378738322196706013428557679949358718247570924314917636028125000000000

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,2

COMMENTS

Product_{j>=1, k>=1} (1 + 1/(j^3*k^3)) = 3.07044599622955113359633939413741321690850038945774000273914990604256664558...

LINKS

Table of n, a(n) for n=0..6.

FORMULA

a(n) ~ c * 4^n * Pi^(2*n) * n^(2*n*(2*n+1)) / exp(4*n^2), where c = 14.2467190172413789737182639605567415110439648274273645215657580983939589... = exp(1/3) * Product_{j>=1, k>=1} (1 + 1/(j^2*k^2)). - Vaclav Kotesovec, Mar 28 2019

MAPLE

a:= n-> mul(mul((i*j)^2+1, i=1..n), j=1..n):

seq(a(n), n=0..7); # Alois P. Heinz, Jun 24 2023

MATHEMATICA

Table[Product[j^2*k^2 + 1, {j, 1, n}, {k, 1, n}], {n, 1, 8}]

Round[Table[Product[k^(1 + 2*n) * Gamma[1 - I/k + n] * Gamma[1 + I/k + n] * Sinh[Pi/k]/Pi, {k, 1, n}], {n, 1, 8}]]

CROSSREFS

Cf. A306760, A324590.

KEYWORD

nonn

AUTHOR

Vaclav Kotesovec, Mar 09 2019

EXTENSIONS

a(0)=1 prepended by Alois P. Heinz, Jun 24 2023

STATUS

approved

A324590

a(n) = n!^(4*n) * Product_{k=1..n} binomial(n + 1/k^2, n).

+10
2

1, 2, 1080, 16133644800, 139256878046022696960000, 6288402750181849898716908922601472000000000, 8322157105451357856813375261666887975745751468393837363200000000000000

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,2

LINKS

Table of n, a(n) for n=0..6.

FORMULA

a(n) ~ n!^(4*n) * n^(Pi^2/6) / A303670.

a(n) ~ n^(4*n^2 + 2*n + Pi^2/6) * (2*Pi)^(2*n) / exp(4*n^2 - 1/3 - gamma*Pi^2/6 + c), where gamma is the Euler-Mascheroni constant A001620 and c = A306774 = Sum_{k>=2} (-1)^k * Zeta(k) * Zeta(2*k) / k.

a(n) = n!^n * A324596(n).

MAPLE

a:= n-> n!^(4*n)*mul(binomial(n+1/k^2, n), k=1..n):

seq(a(n), n=0..7); # Alois P. Heinz, Jun 24 2023

MATHEMATICA

Table[n!^(4*n) * Product[Binomial[1/k^2 + n, n], {k, 1, n}], {n, 1, 8}]

CROSSREFS

Cf. A303670, A306760, A306774, A324589.

KEYWORD

nonn

AUTHOR

Vaclav Kotesovec, Mar 09 2019

EXTENSIONS

a(0)=1 prepended by Alois P. Heinz, Jun 24 2023

STATUS

approved

A324596

a(n) = n!^(3*n) * Product_{k=1..n} binomial(n + 1/k^2, n).

+10
2

1, 2, 270, 74692800, 419731620267960000, 252716802910471719823692648960000, 59736659298524125157504488525739821430187940800000000, 16079377413231597423103950774423398920424350187193326745026311068057600000000000

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,2

LINKS

Table of n, a(n) for n=0..7.

FORMULA

a(n) ~ n!^(3*n) * n^(Pi^2/6) / A303670.

a(n) ~ n^(3*n*(2*n+1)/2 + Pi^2/6) * (2*Pi)^(3*n/2) / exp(3*n^2 - 1/4 - gamma*Pi^2/6 + c), where gamma is the Euler-Mascheroni constant A001620 and c = A306774 = Sum_{k>=2} (-1)^k * Zeta(k) * Zeta(2*k) / k.

MAPLE

a:= n-> n!^(3*n)*mul(binomial(n+1/k^2, n), k=1..n):

seq(a(n), n=0..7); # Alois P. Heinz, Jun 24 2023

MATHEMATICA

Table[n!^(3*n) * Product[Binomial[n + 1/k^2, n], {k, 1, n}], {n, 1, 8}]

CROSSREFS

Cf. A303670, A306760, A306774, A324589, A324597.

KEYWORD

nonn

AUTHOR

Vaclav Kotesovec, Mar 09 2019

EXTENSIONS

a(0)=1 prepended by Alois P. Heinz, Jun 24 2023

STATUS

approved

A324597

a(n) = n!^(4*n) * Product_{k=1..n} binomial(n + 1/k^3, n).

+10
2

1, 2, 918, 11592504000, 86712397842439769400000, 3472997049383321958747830928094241894400000, 4152034082374349458781848863476555783741415883758270213129361920000000

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,2

COMMENTS

In general, for m > 1, Product_{k=1..n} binomial(n + 1/k^m, n) ~ n^Zeta(m) / c(m), where c(m) = Product_{j>=1} Gamma(1 + 1/j^m)).

Equivalently, c(m) = -gamma * Zeta(m) + Sum_{k>=2} (-1)^k*Zeta(k)*Zeta(m*k)/k, where gamma is the Euler-Mascheroni constant A001620.

LINKS

Table of n, a(n) for n=0..6.

FORMULA

a(n) ~ n!^(4*n) * n^Zeta(3) / (Product_{j>=1} Gamma(1 + 1/j^3)).

a(n) ~ n^(4*n^2 + 2*n + Zeta(3)) * (2*Pi)^(2*n) / exp(4*n^2 - 1/3 - gamma*Zeta(3) + c), where c = A306778 = Sum_{k>=2} (-1)^k*Zeta(k)*Zeta(3*k)/k.

MAPLE

a:= n-> n!^(4*n)*mul(binomial(n+1/k^3, n), k=1..n):

seq(a(n), n=0..7); # Alois P. Heinz, Jun 24 2023

MATHEMATICA

Table[n!^(4*n) * Product[Binomial[n + 1/j^3, n], {j, 1, n}], {n, 1, 8}]

CROSSREFS

Cf. A306760, A324596, A306778.

KEYWORD

nonn

AUTHOR

Vaclav Kotesovec, Mar 09 2019

EXTENSIONS

a(0)=1 prepended by Alois P. Heinz, Jun 24 2023

STATUS

approved

A325049

a(n) = Product_{i=0..n, j=0..n} (i!*j! + 1).

+10
1

2, 16, 6480, 97287175440, 1106928595945936328906250000, 856337316801926460412829104011102303451051923953906250000

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,1

LINKS

Table of n, a(n) for n=0..5.

FORMULA

a(n) ~ c * (2*Pi)^((n+1)^2) * n^((n+1)*(6*n^2 + 12*n + 5)/6) / (A^(2*n+2) * exp(3*n^3/2 + 7*n^2/2 + 11*n/6 - 1/3)), where c = Product_{i>=0, j>=0} (1 + 1/(i!*j!)) = 297.557220207478770881166673701943476275955597334672817171839377... and A is the Glaisher-Kinkelin constant A074962.

MATHEMATICA

Table[Product[i!*j! + 1, {i, 0, n}, {j, 0, n}], {n, 0, 7}]

Table[BarnesG[n+2]^(2*n+2) * Product[1 + 1/(i!*j!), {i, 0, n}, {j, 0, n}], {n, 0, 7}]

PROG

(Python)

from math import prod, factorial as f

def a(n): return prod(f(i)*f(j)+1 for i in range(n) for j in range(n))

print([a(n) for n in range(1, 7)]) # Michael S. Branicky, Feb 16 2021

CROSSREFS

Cf. A306760, A325051.

KEYWORD

nonn

AUTHOR

Vaclav Kotesovec, Mar 26 2019

STATUS

approved