A084448 -id:A084448 - cp's OEIS frontend

A258351 Expansion of Product_{k>=1} 1/(1-x^k)^(k(k-1)(k-2)).

1, 0, 0, 6, 24, 60, 141, 354, 996, 2720, 7194, 18306, 46154, 115506, 288195, 713210, 1749732, 4253148, 10259302, 24573390, 58491312, 138371354, 325415727, 760899396, 1769420183, 4093054602, 9420739965, 21578842582, 49199229066, 111672215658, 252381169048

Offset: 0

Vaclav Kotesovec, May 27 2015

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..1000

Cf. A000335, A023872, A258345, A258347, A258348, A258349, A258350, A258352.

nmax=40; CoefficientList[Series[Product[1/(1-x^k)^(k*(k-1)*(k-2)),{k,1,nmax}],{x,0,nmax}],x]

a(n) ~ (3*Zeta(5))^(79/600) / (2^(21/200) * sqrt(5*Pi) * n^(379/600)) * exp(2*Zeta'(-1) + 3*Zeta(3)/(4*Pi^2) - Pi^16 / (518400000 * Zeta(5)^3) + Pi^8 * Zeta(3) / (36000 * Zeta(5)^2) - Zeta(3)^2 / (15*Zeta(5)) + Zeta'(-3) + (-Pi^12 / (1800000 * 2^(3/5) * 3^(1/5) * Zeta(5)^(11/5)) + Pi^4 * Zeta(3) / (150 * 2^(3/5) * 3^(1/5) * Zeta(5)^(6/5))) * n^(1/5) + (-Pi^8 / (12000 * 2^(1/5) * 3^(2/5) * Zeta(5)^(7/5)) + Zeta(3) / (2^(1/5) * (3*Zeta(5))^(2/5))) * n^(2/5) - Pi^4 / (30 * 2^(4/5) * (3*Zeta(5))^(3/5)) * n^(3/5) + 5 * (3*Zeta(5))^(1/5) / 2^(7/5) * n^(4/5)), where Zeta(3) = A002117, Zeta(5) = A013663, Zeta'(-1) = A084448 = 1/12 - log(A074962), Zeta'(-3) = ((gamma + log(2*Pi) - 11/6)/30 - 3*Zeta'(4)/Pi^4)/4.

A263030 Decimal expansion of a constant related to A262876 and A262946 (negated).

Original entry on oeis.org

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

Offset: 0

Vaclav Kotesovec, Oct 08 2015

			-0.18870819197952853237641009864920797359211446726842922150941743378232...

Cf. A262876, A262877, A262946, A262947, A263031, A075700, A084448, A263406, A263415.

NIntegrate[1/x*(Exp[-2*x]/(1 - Exp[-3*x])^2 - 1/(9*x^2) - 1/(9*x) + Exp[-x]/36), {x, 0, Infinity}, WorkingPrecision -> 120, MaxRecursion -> 100, PrecisionGoal -> 110]

Integral_{x=0..infinity} 1/x*(exp(-2*x)/(1 - exp(-3*x))^2 - 1/(9*x^2) - 1/(9*x) + exp(-x)/36) dx.

exp(3*(A263030+A263031)) = A^2 * Gamma(1/3) / (3^(11/12) * exp(1/6) * sqrt(2*Pi)), where A = A074962 is the Glaisher-Kinkelin constant.

A263031 Decimal expansion of a constant related to A262877 and A262947 (negated).

Original entry on oeis.org

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

Offset: 0

Vaclav Kotesovec, Oct 08 2015

			-0.01453742918328403360502029450226209036054149759346444138152247405534...

Cf. A262876, A262877, A262946, A262947, A263030, A075700, A084448, A263405, A263414.

NIntegrate[1/x*(Exp[-x]/(1 - Exp[-3*x])^2 - 1/(9*x^2) - 2/(9*x) - 5*Exp[-x]/36), {x, 0, Infinity}, WorkingPrecision -> 120, MaxRecursion -> 100, PrecisionGoal -> 110]

Integral_{x=0..infinity} 1/x*(exp(-x)/(1 - exp(-3*x))^2 - 1/(9*x^2) - 2/(9*x) - 5*exp(-x)/36) dx.

exp(3*(A263030+A263031)) = A^2 * Gamma(1/3) / (3^(11/12) * exp(1/6) * sqrt(2*Pi)), where A = A074962 is the Glaisher-Kinkelin constant.

A263358 Expansion of Product_{k>=1} 1/(1-x^(k+2))^k.

Original entry on oeis.org

1, 0, 0, 1, 2, 3, 5, 7, 12, 18, 29, 43, 69, 101, 155, 231, 347, 509, 759, 1106, 1626, 2359, 3428, 4938, 7127, 10194, 14587, 20756, 29498, 41716, 58932, 82888, 116413, 162924, 227602, 316988, 440696, 610953, 845469, 1167118, 1608178, 2210888, 3034124, 4155111

Offset: 0

Vaclav Kotesovec, Oct 16 2015

nonn

In general, if v>=0 and g.f. = Product_{k>=1} 1/(1-x^(k+v))^k, then a(n) ~ d1(v) * n^(v^2/6 - 25/36) * exp(-Pi^4 * v^2 / (432*Zeta(3)) + 3*Zeta(3)^(1/3) * n^(2/3)/2^(2/3) - v * Pi^2 * n^(1/3) / (3 * 2^(4/3) * Zeta(3)^(1/3))) / (sqrt(3*Pi) * 2^(v^2/6 + 11/36) * Zeta(3)^(v^2/6 - 7/36)), where Zeta(3) = A002117.
d1(v) = exp(Integral_{x=0..infinity} (1/(x*exp((v-1)*x) * (exp(x)-1)^2) - (6*v^2-1) / (12*x*exp(x)) + v/x^2 - 1/x^3) dx).
d1(v) = (exp(Zeta'(-1) - v*Zeta'(0))) / Product_{j=0..v-1} j!, where Zeta'(0) = -A075700, Zeta'(-1) = A084448 and Product_{j=0..v-1} j! = A000178(v-1).
d1(v) = exp(1/12) * (2*Pi)^(v/2) / (A * G(v+1)), where A = A074962 is the Glaisher-Kinkelin constant and G is the Barnes G-function.

Vaclav Kotesovec, Table of n, a(n) for n = 0..5000
Vaclav Kotesovec, A method of finding the asymptotics of q-series based on the convolution of generating functions, arXiv:1509.08708 [math.CO], Sep 30 2015
Vaclav Kotesovec, Graph - The asymptotic ratio (30000 terms)
Eric Weisstein's World of Mathematics, Barnes G-Function

Cf. A000219, A052847, A263359, A263360, A263361, A263362, A263363, A263364.

with(numtheory):
a:= proc(n) option remember; `if`(n=0, 1, add(add(d*
      max(0, d-2), d=divisors(j))*a(n-j), j=1..n)/n)
    end:
seq(a(n), n=0..50);  # Alois P. Heinz, Oct 16 2015

nmax = 50; CoefficientList[Series[Product[1/(1-x^(k+2))^k, {k, 1, nmax}], {x, 0, nmax}], x]
nmax = 50; CoefficientList[Series[E^Sum[x^(3*k)/(k*(1-x^k)^2), {k, 1, nmax}], {x, 0, nmax}], x]

G.f.: exp(Sum_{k>=1} x^(3*k)/(k*(1-x^k)^2)).

a(n) ~ exp(1/12 - Pi^4/(108*Zeta(3)) - Pi^2 * n^(1/3) / (3 * 2^(1/3) * Zeta(3)^(1/3)) + 3 * 2^(-2/3) * Zeta(3)^(1/3) * n^(2/3)) * 2^(1/36) * sqrt(Pi) / (A * sqrt(3) * Zeta(3)^(17/36) * n^(1/36)), where Zeta(3) = A002117 and A = A074962 is the Glaisher-Kinkelin constant.

A263364 Expansion of Product_{k>=1} 1/(1-x^(k+8))^k.

Original entry on oeis.org

1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 13, 18, 23, 33, 43, 60, 77, 103, 130, 168, 209, 267, 331, 420, 526, 667, 839, 1069, 1347, 1711, 2160, 2733, 3437, 4336, 5435, 6828, 8543, 10699, 13357, 16703, 20820, 25986, 32362, 40327, 50152, 62407

Offset: 0

Vaclav Kotesovec, Oct 16 2015

nonn

In general, if v>=0 and g.f. = Product_{k>=1} 1/(1-x^(k+v))^k, then a(n) ~ d1(v) * n^(v^2/6 - 25/36) * exp(-Pi^4 * v^2 / (432*Zeta(3)) + 3*Zeta(3)^(1/3) * n^(2/3)/2^(2/3) - v * Pi^2 * n^(1/3) / (3 * 2^(4/3) * Zeta(3)^(1/3))) / (sqrt(3*Pi) * 2^(v^2/6 + 11/36) * Zeta(3)^(v^2/6 - 7/36)), where Zeta(3) = A002117.
d1(v) = exp(Integral_{x=0..infinity} (1/(x*exp((v-1)*x) * (exp(x)-1)^2) - (6*v^2-1) / (12*x*exp(x)) + v/x^2 - 1/x^3) dx).
d1(v) = (exp(Zeta'(-1) - v*Zeta'(0))) / Product_{j=0..v-1} j!, where Zeta'(0) = -A075700, Zeta'(-1) = A084448 and Product_{j=0..v-1} j! = A000178(v-1).
d1(v) = exp(1/12) * (2*Pi)^(v/2) / (A * G(v+1)), where A = A074962 is the Glaisher-Kinkelin constant and G is the Barnes G-function.

Vaclav Kotesovec, Table of n, a(n) for n = 0..5000
Vaclav Kotesovec, A method of finding the asymptotics of q-series based on the convolution of generating functions, arXiv:1509.08708 [math.CO], Sep 30 2015
Eric Weisstein's World of Mathematics, Barnes G-Function.

Cf. A000219 (v=0), A052847 (v=1), A263358 (v=2), A263359 (v=3), A263360 (v=4), A263361 (v=5), A263362 (v=6), A263363 (v=7).

with(numtheory):
a:= proc(n) option remember; `if`(n=0, 1, add(add(d*
      max(0, d-8), d=divisors(j))*a(n-j), j=1..n)/n)
    end:
seq(a(n), n=0..60);  # Alois P. Heinz, Oct 16 2015

nmax = 60; CoefficientList[Series[Product[1/(1-x^(k+8))^k, {k, 1, nmax}], {x, 0, nmax}], x]
nmax = 60; CoefficientList[Series[E^Sum[x^(9*k)/(k*(1-x^k)^2), {k, 1, nmax}], {x, 0, nmax}], x]

G.f.: exp(Sum_{k>=1} x^(9*k)/(k*(1-x^k)^2)).

a(n) ~ exp(1/12 - 4*Pi^4/(27*Zeta(3)) - 2^(5/3) * Pi^2 * n^(1/3) / (3 * Zeta(3)^(1/3)) + 3 * 2^(-2/3) * Zeta(3)^(1/3) * n^(2/3)) * n^(359/36) * Pi^(7/2) / (8026324992000 * A * 2^(35/36) * sqrt(3) * Zeta(3)^(377/36)), where Zeta(3) = A002117 and A = A074962 is the Glaisher-Kinkelin constant.

A225746 Decimal expansion of the logarithm of Glaisher's constant.

Original entry on oeis.org

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

Offset: 1

Jean-François Alcover, May 14 2013

			0.248754477033784262547252993576113976097369713668535116999855639690693032999...

Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 2.15 Glaisher-Kinkelin constant, p. 135.

Jesús Guillera and Jonathan Sondow, Double integrals and infinite products for some classical constants via analytic continuations of Lerch's transcendent, The Ramanujan Journal, Vol. 16, No. 3 (2008), pp. 247-270; arXiv preprint, arXiv:math/0506319 [math.NT], 2005-2006.
Jean-Christophe Pain, Two integral representations for the logarithm of the Glaisher-Kinkelin constant, arXiv:2405.05264 [math.GM], 2024.
Eric Weisstein's World of Mathematics, Glaisher-Kinkelin Constant.

Cf. A001620, A073002, A074962, A084448.

RealDigits[Log[Glaisher], 10, 100] // First

1/12-zeta'(-1) \\ Charles R Greathouse IV, Dec 12 2013

Equals 1/12 - zeta'(-1).
Also equals (gamma + log(2*Pi))/12 -zeta'(2)/(2*Pi^2).
From Amiram Eldar, Apr 15 2021: (Start)
Equals lim_{n->oo} (Sum_{k=1..n} k*log(k) - (n^2/2 + n/2 + 1/12)*log(n) + n^2/4).
Equals 1/8 + (1/2) * Sum_{n>=0} ((1/(n+1)) * Sum_{k=0..n} (-1)^(k+1) * binomial(n,k) * (k+1)^2 * log(k+1)) (Guillera and Sondow, 2008). (End)

A263352 Expansion of Product_{k>=1} 1/(1 - x^(2*k+3))^k.

Original entry on oeis.org

1, 0, 0, 0, 0, 1, 0, 2, 0, 3, 1, 4, 2, 5, 6, 7, 10, 9, 19, 14, 29, 23, 46, 38, 66, 64, 99, 107, 143, 171, 211, 270, 311, 418, 465, 633, 698, 945, 1049, 1399, 1579, 2052, 2364, 2997, 3527, 4366, 5219, 6339, 7686, 9197, 11234, 13321, 16340, 19261, 23622, 27796

Offset: 0

Vaclav Kotesovec, Oct 16 2015

nonn

From Vaclav Kotesovec, Oct 17 2015: (Start)
In general, if g.f. = Product_{k>=1} 1/(1-x^(2*k+v))^k and v>0 is odd, then a(n) ~ d2(v) * (2*n)^(v^2/24 - 25/36) * exp(-Pi^4 * v^2 / (1728*Zeta(3)) - Pi^2 * v * n^(1/3) /(3 * 2^(8/3) * Zeta(3)^(1/3)) + 3*Zeta(3)^(1/3) * n^(2/3) / 2^(4/3)) / (sqrt(3*Pi) * Zeta(3)^(v^2/24 - 7/36)), where Zeta(3) = A002117.
d2(v) = exp(Integral_{x=0..infinity} 1/(x*exp((v-2)*x) * (exp(2*x)- 1)^2) - (3*v^2-2)/(24*x*exp(x)) + v/(4*x^2) - 1/(4*x^3) dx).
d2(v) = 2^(v/4 - 1/12) * exp(-Zeta'(-1)/2) / Product_{j=1..(v-1)/2} (2*j-1)!!, where Zeta'(-1) = A084448 and Product_{j=1..(v-1)/2} (2*j-1)!! = A057863((v-1)/2).
d2(v) = 2^(1/12 + v/4 - v^2/8) * exp(1/12) * Pi^(v/4) / (A * G(v/2 + 1)), where A = A074962 is the Glaisher-Kinkelin constant and G is the Barnes G-function.
(End)

Vaclav Kotesovec, Table of n, a(n) for n = 0..1000
Vaclav Kotesovec, A method of finding the asymptotics of q-series based on the convolution of generating functions, arXiv:1509.08708 [math.CO], Sep 30 2015
Eric Weisstein's World of Mathematics, Barnes G-Function.

Cf. A052847, A000219, A263150, A035528, A263395, A263396, A263397.

with(numtheory):
a:= proc(n) option remember; `if`(n=0, 1, add(add(d*
      `if`(d>1 and d::odd, (d-3)/2, 0),
      d=divisors(j))*a(n-j), j=1..n)/n)
    end:
seq(a(n), n=0..60);  # Alois P. Heinz, Oct 17 2015

nmax = 60; CoefficientList[Series[Product[1/(1 - x^(2*k+3))^k, {k, 1, nmax}], {x, 0, nmax}], x]

a(n) ~ 2^(25/72) * sqrt(A) * exp(-1/24 + 3 * 2^(-4/3) * Zeta(3)^(1/3) * n^(2/3) - Pi^4/(192*Zeta(3)) - Pi^2 * n^(1/3)/(2^(8/3) * Zeta(3)^(1/3))) / (sqrt(3*Pi) * Zeta(3)^(13/72) * n^(23/72)), where Zeta(3) = A002117 and A = A074962 is the Glaisher-Kinkelin constant.

A213080 Decimal expansion of Product_{n>=1} n! /(sqrt(2Pin) * (n/e)^n * (1+1/n)^(1/12)).

Original entry on oeis.org

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

Offset: 1

Michael David Hirschhorn, Jun 04 2012

Just as Stirling's formula for the asymptotic expansion of n! involves the constant sqrt{2 Pi}, the asymptotic expansion of the product of all binomial coefficients in a row of Pascal's triangle involves a constant, the reciprocal of the constant C defined and evaluated here.
From Bernd C. Kellner, Oct 13 2024: (Start)
It turns out that 1/C is not the complete asymptotic constant for the product of the binomial coefficients in a row of Pascal's triangle. A constant factor of (2*Pi)^(-1/4) was overlooked in the asymptotic expansion of that product given by Hirschhorn in 2013. The correct asymptotic constant is A377023.
However, the constant C equals the constant F(1) as introduced before in Kellner 2009. The constants F(1), F(2), ... occur in the same context of asymptotic constants related to asymptotic products of factorials as well as of binomial and multinomial coefficients. Moreover, the sequence (F(k)){k >= 1} is strictly decreasing with limit 1. For example, for k >= 1 the asymptotic product Prod{v >= 1} (k*v)! has the asymptotic constant F(k)*A^k*(2*Pi)^(1/4), where A = A074962 denotes the Glaisher-Kinkelin constant. Let gamma = A001620 be Euler's constant and Gamma(x) be the gamma function.
For k >= 1, the constants F(k) can be computed by an explicit formula and a divergent series expansion, as follows. We have log(F(k)) = (1/(12*k))*(1-log(k)) + (k/4)*log(2*Pi) - ((k^2+1)/k)*log(A) - Sum_{v=1..k-1} (v/k)*log(Gamma(v/k)) = gamma/(12*k) - t*zeta(3)/(360*k^3) with some t in (0,1), respectively.
It follows that log(F(1)) = 1/12 + log(2*Pi)/4 - 2*log(A) = gamma/12 - t*zeta(3)/360 with some t in (0,1), and so F(1) lies in the interval (1.0457...,1.0492...) (see Kellner 2009 and 2024). (End)

			1.04633506677050318098095065697776037101974218113264442441587534042035751563744...

G. C. Greubel, Table of n, a(n) for n = 1..10000
Michael D. Hirschhorn, On the asymptotic behavior of Product_{k=0..n} C(n,k), Fib. Q., 51 (2013), 163-173.
Bernd C. Kellner, On asymptotic constants related to products of Bernoulli numbers and factorials, Integers 9 (2009), Article #A08, 83-106; alternative link; arXiv:0604505 [math.NT], 2006.
Bernd C. Kellner, Asymptotic products of binomial and multinomial coefficients revisited, Integers 24 (2024), Article #A59, 10 pp.; arXiv:2312.11369 [math.CO], 2023.

Cf. A074962, A000178, A084448, A241140, A272097.

Cf. A001620, A002117, A377023, A377024.

exp(2*Zeta(1,-1)-1/12)*(2*Pi)^(1/4); evalf(%,100); # Peter Luschny, Jun 22 2012

RealDigits[(Exp[1]^(1/12) (2 Pi)^(1/4))/Glaisher^2, 10, 100][[1]] (*Peter Luschny, Jun 22 2012 *)

exp(2*zeta'(-1)-1/12)*(2*Pi)^(1/4) \\ Charles R Greathouse IV, Dec 12 2013

import mpmath
mpmath.mp.pretty=True; mpmath.mp.dps = 200 #precision
mpmath.exp(2*mpmath.zeta(-1,1,1)-1/12)*(2*pi)^(1/4) # Peter Luschny, Jun 22 2012

Equals (exp(1)^(1/12)*(2*Pi)^(1/4))/A^2 where A denotes the Glaisher-Kinkelin constant.
Equals exp(2*zeta'(-1)-1/12)*(2*Pi)^(1/4).
A closely related constant is K = Product_{n>=1} (n!*(e/n)^(n+1/2))/ ((1+1/(n+1/2))^(1/12)*sqrt(2*Pi*e)) = (2^(1/6)*(3*e)^(1/12)*Pi^(1/4))/A^2 = exp(2*zeta'(-1)-1/12)*2^(1/6)*3^(1/12)*Pi^(1/4) = 1.082293504658977773529439... - Peter Luschny, Jun 22 2012
The sqrt of the constant equals Limit_{n>=1} (Product_{k=1..n-1} k!) / f(n) where f(n) = (2*Pi)^(n/2-1/8)*exp(1/24-3/4*n^2)*n^(1/2*n^2-1/12). - Peter Luschny, Jun 23 2012

A263397 Expansion of Product_{k>=1} 1/(1 - x^(2*k+9))^k.

Original entry on oeis.org

1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 2, 0, 3, 0, 4, 0, 5, 0, 6, 1, 7, 2, 8, 6, 9, 10, 10, 19, 11, 28, 13, 44, 15, 60, 20, 85, 29, 110, 44, 146, 69, 183, 111, 233, 171, 286, 262, 358, 391, 441, 568, 553, 808, 697, 1129, 898, 1543, 1174, 2080, 1563, 2766

Offset: 0

Vaclav Kotesovec, Oct 16 2015

nonn

In general, if g.f. = Product_{k>=1} 1/(1-x^(2*k+v))^k and v>0 is odd, then a(n) ~ d2(v) * (2*n)^(v^2/24 - 25/36) * exp(-Pi^4 * v^2 / (1728*Zeta(3)) - Pi^2 * v * n^(1/3) /(3 * 2^(8/3) * Zeta(3)^(1/3)) + 3*Zeta(3)^(1/3) * n^(2/3) / 2^(4/3)) / (sqrt(3*Pi) * Zeta(3)^(v^2/24 - 7/36)), where Zeta(3) = A002117.
d2(v) = exp(Integral_{x=0..infinity} 1/(x*exp((v-2)*x) * (exp(2*x)- 1)^2) - (3*v^2-2)/(24*x*exp(x)) + v/(4*x^2) - 1/(4*x^3) dx).
d2(v) = 2^(v/4 - 1/12) * exp(-Zeta'(-1)/2) / Product_{j=1..(v-1)/2} (2*j-1)!!, where Zeta'(-1) = A084448 and Product_{j=1..(v-1)/2} (2*j-1)!! = A057863((v-1)/2).
d2(v) = 2^(1/12 + v/4 - v^2/8) * exp(1/12) * Pi^(v/4) / (A * G(v/2 + 1)), where A = A074962 is the Glaisher-Kinkelin constant and G is the Barnes G-function.

Vaclav Kotesovec, Table of n, a(n) for n = 0..5000
Vaclav Kotesovec, A method of finding the asymptotics of q-series based on the convolution of generating functions, arXiv:1509.08708 [math.CO], Sep 30 2015
Eric Weisstein's World of Mathematics, Barnes G-Function.

Cf. A035528 (v=-1), A263150 (v=1), A263352 (v=3), A263395 (v=5), A263396 (v=7).

with(numtheory):
a:= proc(n) option remember; local r; `if`(n=0, 1,
       add(add(`if`(irem(d-8, 2, 'r')=1, d*r, 0)
       , d=divisors(j))*a(n-j), j=1..n)/n)
    end:
seq(a(n), n=0..65);  # Alois P. Heinz, Oct 17 2015

nmax = 60; CoefficientList[Series[Product[1/(1 - x^(2*k+9))^k, {k, 1, nmax}], {x, 0, nmax}], x]
nmax = 60; CoefficientList[Series[E^Sum[x^(11*k)/(k*(1-x^(2*k))^2), {k, 1, nmax}], {x, 0, nmax}], x]

G.f.: exp(Sum_{k>=1} x^(11*k)/(k*(1-x^(2*k))^2)).

a(n) ~ 16 * 2^(61/72) * exp(-1/24 - 3*Pi^4/(64*Zeta(3)) - 3*Pi^2 * n^(1/3) / (2^(8/3) * Zeta(3)^(1/3)) + 3 * Zeta(3)^(1/3) * n^(2/3) / 2^(4/3)) * sqrt(A) * n^(193/72) / (4725*sqrt(3*Pi) * Zeta(3)^(229/72)), where Zeta(3) = A002117 and A = A074962 is the Glaisher-Kinkelin constant.

A249386 Decimal expansion of the constant 'a' appearing in the asymptotic expression of the number of plane partitions of n as an^(-25/36)exp(b*n^(2/3)).

Original entry on oeis.org

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

Offset: 0

Jean-François Alcover, Oct 27 2014

The paper by Finch contains an error: the denominator should be sqrt(3*Pi), not sqrt(Pi). The constant 0.4009988836 is wrong. The formula in A000219 and the article by L. Mutafchiev and E. Kamenov (page 6) is correct. - Vaclav Kotesovec, Oct 27 2014. [In new version of prt.pdf is already corrected. - Vaclav Kotesovec, May 11 2015]

			0.231516813448898370560356406406332110855129212593287926597944524...

G. C. Greubel, Table of n, a(n) for n = 0..5000
Steven Finch, Integer Partitions, September 22, 2004. [Cached copy, with permission of the author]
L. Mutafchiev and E. Kamenov, On The Asymptotic Formula for the Number of Plane Partitions of Positive Integers, arXiv:math/0601253 [math.CO], 2006; C. R. Acad. Bulgare Sci. 59(2006), No. 4, 361-366.

Cf. A000219, A020805, A084448, A239049.

a = Zeta[3]^(7/36)*Exp[Zeta'[-1]]/(2^(11/36)*Sqrt[3*Pi]); RealDigits[a, 10, 104] // First

Equals zeta(3)^(7/36)*exp(zeta'(-1))/(2^(11/36)*sqrt(3*Pi)).

Equals exp(1/12) * A002117^(7/36) / (A074962 * 2^(11/36) * sqrt(3*Pi)). - Vaclav Kotesovec, Mar 02 2015

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