login
Expansion of Product_{k>=1} 1/(1 - x^k)^((5*k-1)*binomial(k+2,3)/4).
3

%I #10 Apr 03 2019 02:59:56

%S 1,1,10,45,185,710,2766,10270,37444,132765,462327,1579563,5311361,

%T 17584084,57414594,185032557,589183035,1854974066,5778722817,

%U 17823440534,54458411332,164917654587,495219323844,1475145786950,4360576440676,12796007418881,37287660835368,107930276062786

%N Expansion of Product_{k>=1} 1/(1 - x^k)^((5*k-1)*binomial(k+2,3)/4).

%C Euler transform of A002418.

%H N. J. A. Sloane, <a href="/transforms.txt">Transforms</a>

%F G.f.: Product_{k>=1} 1/(1 - x^k)^A002418(k).

%F G.f.: exp(Sum_{k>=1} x^k*(1 + 4*x^k)/(k*(1 - x^k)^5)).

%F a(n) ~ (5/7)^(703/8640)/(2 * 3^(2143/2880) * n^(5023/8640) * Pi^(17/1440)) * exp(-1/144 + (1/12-Zeta'(-1))/12 - (21 * Zeta(3))/(400 * Pi^2) + (62921 * Zeta(5))/(80000 * Pi^4) - (194481 * Zeta(3) * Zeta(5)^2)/(50 * Pi^12) - (200120949 * Zeta(5)^3)/(1250 * Pi^14) + (28594081676916 * Zeta(5)^5)/(3125 * Pi^24) + (7 * Zeta'(-3))/12 + ((-343 * (7/5)^(1/6) * Pi)/(96000 * sqrt(3)) + (147 * (7/5)^(1/6) * sqrt(3) * Zeta(3) * Zeta(5))/(10 * Pi^7) + (1058841 * (7/5)^(1/6) * sqrt(3) * Zeta(5)^2)/(2000 * Pi^9) - (18211006359 * (7/5)^(1/6) * sqrt(3) * Zeta(5)^4)/(500 * Pi^19)) * n^(1/6) + (-((7/5)^(1/3) * Zeta(3))/(4 * Pi^2) - (1029 * (7/5)^(1/3) * Zeta(5))/(200 * Pi^4) + (10890936 * (7/5)^(1/3) * Zeta(5)^3)/(25 * Pi^14)) * n^(1/3) + ((7 * sqrt(7/15) * Pi)/120 - (9261 * sqrt(21/5) * Zeta(5)^2)/(5 * Pi^9)) * sqrt(n) + ((63 * (7/5)^(2/3) * Zeta(5))/(2 * Pi^4)) * n^(2/3) + ((2 * sqrt(3) * Pi)/(5^(5/6) * 7^(1/6))) * n^(5/6)). - _Vaclav Kotesovec_, Jul 28 2018

%p a:=series(mul(1/(1-x^k)^((5*k-1)*binomial(k+2,3)/4),k=1..100),x=0,28): seq(coeff(a,x,n),n=0..27); # _Paolo P. Lava_, Apr 02 2019

%t nmax = 27; CoefficientList[Series[Product[1/(1 - x^k)^((5 k - 1) Binomial[k + 2, 3]/4), {k, 1, nmax}], {x, 0, nmax}], x]

%t nmax = 27; CoefficientList[Series[Exp[Sum[x^k (1 + 4 x^k)/(k (1 - x^k)^5), {k, 1, nmax}]], {x, 0, nmax}], x]

%t a[n_] := a[n] = If[n == 0, 1, Sum[Sum[d^2 (d + 1) (d + 2) (5 d - 1)/24, {d, Divisors[k]}] a[n - k], {k, 1, n}]/n]; Table[a[n], {n, 0, 27}]

%Y Cf. A000391, A002418, A278769, A279218, A305653, A317017, A317019, A317021.

%K nonn

%O 0,3

%A _Ilya Gutkovskiy_, Jul 19 2018