A317021 - OEIS

A317021

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

1, 1, 11, 51, 216, 861, 3477, 13367, 50377, 184667, 664484, 2345230, 8142476, 27825576, 93750686, 311682789, 1023547782, 3322634928, 10669887669, 33916213669, 106776876109, 333111724130, 1030264525744, 3160359629535, 9618807643826, 29057370625281, 87153154537437

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

OFFSET

0,3

COMMENTS

Euler transform of A002419.

LINKS

Table of n, a(n) for n=0..26.

N. J. A. Sloane, Transforms

FORMULA

G.f.: Product_{k>=1} 1/(1 - x^k)^A002419(k).

G.f.: exp(Sum_{k>=1} x^k*(1 + 5*x^k)/(k*(1 - x^k)^5)).

a(n) ~ 1/(2^(1987/2160) * 3^(713/1080) * 7^(173/2160) * n^(1253/2160) * Pi^(7/360)) * exp(-1/72 + (1/12-Zeta'(-1))/6 - Zeta(3)/(30 * Pi^2) + (111 * Zeta(5))/(200 * Pi^4) - (7056 * Zeta(3) * Zeta(5)^2)/Pi^12 - (592704 * Zeta(5)^3)/(5 * Pi^14) + (43016085504 * Zeta(5)^5)/(5 * Pi^24) + (2 * Zeta'(-3))/3 + ((-7 * (7/2)^(1/6) * Pi)/(3200 * 3^(2/3)) + (14 * 2^(5/6) * 3^(1/3) * 7^(1/6) * Zeta(3) * Zeta(5))/Pi^7 + (1029 * 2^(5/6) * 3^(1/3) * 7^(1/6) * Zeta(5)^2)/(5 * Pi^9) - (17978688 * 2^(5/6) * 3^(1/3) * 7^(1/6) * Zeta(5)^4)/Pi^19) * n^(1/6) + (-((7/6)^(1/3) * Zeta(3))/(2 * Pi^2) - (7 * 3^(2/3) * (7/2)^(1/3) * Zeta(5))/(5 * Pi^4) + (75264 * 6^(2/3) * 7^(1/3) * Zeta(5)^3)/Pi^14) * n^(1/3) + ((sqrt(7/2) * Pi)/60 - (1008 * sqrt(14) * Zeta(5)^2)/Pi^9) * sqrt(n) + ((6 * 6^(1/3) * 7^(2/3) * Zeta(5))/Pi^4) * n^(2/3) + ((2 * (2/7)^(1/6) * 3^(2/3) * Pi)/5) * n^(5/6)). - Vaclav Kotesovec, Jul 28 2018

MAPLE

a:= proc(n) option remember; `if`(n=0, 1, add(add(

(3*d-1)*binomial(d+2, 3)/2*d, d=numtheory

[divisors](j))*a(n-j), j=1..n)/n)

end:

seq(a(n), n=0..25); # Alois P. Heinz, Jul 19 2018

MATHEMATICA

nmax = 26; CoefficientList[Series[Product[1/(1 - x^k)^((3 k - 1) Binomial[k + 2, 3]/2), {k, 1, nmax}], {x, 0, nmax}], x]

nmax = 26; CoefficientList[Series[Exp[Sum[x^k (1 + 5 x^k)/(k (1 - x^k)^5), {k, 1, nmax}]], {x, 0, nmax}], x]

a[n_] := a[n] = If[n == 0, 1, Sum[Sum[d^2 (d + 1) (d + 2) (3 d - 1)/12, {d, Divisors[k]}] a[n - k], {k, 1, n}]/n]; Table[a[n], {n, 0, 26}]

CROSSREFS

Cf. A000391, A002419, A274998, A279219, A305653, A317017, A317019, A317020.