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A258343 Expansion of Product_{k>=1} (1+x^k)^(k*(k+1)*(k+2)/6).

%I A258343 #11 May 28 2018 19:43:52
%S A258343 1,1,4,14,36,101,260,669,1669,4116,9932,23636,55483,128532,294422,
%T A258343 667026,1496232,3324720,7323570,15998749,34679966,74622839,159454379,
%U A258343 338472749,713956569,1496950669,3120663129,6469901522,13343153563,27379250529,55907749171
%N A258343 Expansion of Product_{k>=1} (1+x^k)^(k*(k+1)*(k+2)/6).
%H A258343 Vaclav Kotesovec, <a href="/A258343/b258343.txt">Table of n, a(n) for n = 0..1000</a>
%F A258343 a(n) ~ (3*Zeta(5))^(1/10) / (2^(523/720) * 5^(2/5) * sqrt(Pi) * n^(3/5)) * exp(-2401 * Pi^16 / (10497600000000 * Zeta(5)^3) + 49*Pi^8 * Zeta(3) / (16200000 * Zeta(5)^2) - Zeta(3)^2 / (150*Zeta(5)) + (343*Pi^12 / (2430000000 * 2^(3/5) * 15^(1/5) * Zeta(5)^(11/5)) - 7*Pi^4 * Zeta(3) / (4500 * 2^(3/5) * 15^(1/5) * Zeta(5)^(6/5))) * n^(1/5) + (-49*Pi^8 / (1080000 * 2^(1/5) * 15^(2/5) * Zeta(5)^(7/5)) + Zeta(3) / (2^(6/5) * (15*Zeta(5))^(2/5))) * n^(2/5) + 7*Pi^4 / (180 * 2^(4/5) * (15*Zeta(5))^(3/5)) * n^(3/5) + 5*(15*Zeta(5))^(1/5) / 2^(12/5) * n^(4/5)), where Zeta(3) = A002117, Zeta(5) = A013663.
%F A258343 G.f.: exp(Sum_{k>=1} (-1)^(k+1)*x^k/(k*(1 - x^k)^4)). - _Ilya Gutkovskiy_, May 28 2018
%p A258343 b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0, add(
%p A258343       binomial(binomial(i+2, 3), j)*b(n-i*j, i-1), j=0..n/i)))
%p A258343     end:
%p A258343 a:= n-> b(n$2):
%p A258343 seq(a(n), n=0..30);  # _Alois P. Heinz_, May 28 2018
%t A258343 nmax=40; CoefficientList[Series[Product[(1+x^k)^(k*(k+1)*(k+2)/6),{k,1,nmax}],{x,0,nmax}],x]
%Y A258343 Cf. A248882, A028377, A258341, A258342, A258344, A258345, A258346.
%K A258343 nonn
%O A258343 0,3
%A A258343 _Vaclav Kotesovec_, May 27 2015