A327050 - OEIS

%I #11 Jul 23 2023 15:52:33

%S 1,2,6,14,32,66,136,260,494,902,1620,2832,4890,8260,13792,22664,36824,

%T 59060,93814,147364,229490,354052,541916,822736,1240292,1856246,

%U 2760368,4078522,5990900,8749052,12708920,18363656,26404386,37783040,53820120,76324576

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

%C Convolution of A327047 and A327044.

%C In general, for fixed m>=1, if g.f. = Product_{k>=1} (Product_{j=1..m} (1 + x^(j*k)) / (1 - x^(j*k))), then a(n) ~ sqrt(Gamma(m+1)) * HarmonicNumber(m)^((m+1)/4) * exp(Pi*sqrt(HarmonicNumber(m)*n)) / (2^(3*(m+1)/2) * n^((m+3)/4)).

%H Vaclav Kotesovec, <a href="/A327050/b327050.txt">Table of n, a(n) for n = 0..10000</a>

%F a(n) ~ 137^(3/2) * exp(sqrt(137*n/15)*Pi/2) / (15*2^(21/2)*n^2).

%t nmax = 50; CoefficientList[Series[Product[(1+x^k) * (1+x^(2*k)) * (1+x^(3*k)) * (1+x^(4*k)) * (1+x^(5*k)) / ((1-x^k) * (1-x^(2*k)) * (1-x^(3*k)) * (1-x^(4*k)) * (1-x^(5*k))), {k, 1, nmax}], {x, 0, nmax}], x]

%t With[{nn=50,xk=x^(k Range[5])},CoefficientList[Series[Product[Times@@(1+xk)/Times@@(1-xk),{k,nn}],{x,0,nn}],x]] (* _Harvey P. Dale_, Jul 23 2023 *)

%Y Cf. A015128, A246584, A327048, A327049.

%Y Cf. A301554.

%K nonn

%O 0,2

%A _Vaclav Kotesovec_, Aug 16 2019