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%I #28 Mar 09 2015 09:49:56
%S 1,5,30,145,660,2777,11160,42805,158490,568050,1980607,6735380,
%T 22402610,73022755,233692345,735350970,2278153310,6956560935,
%U 20958613740,62354061740,183332498533,533074229590,1533842417185,4369816273820,12332669124455,34495668855729
%N G.f.: Product_{j>=1} 1/(1-x^j)^binomial(j+4,4).
%C In general, if g.f. = product_{j>=1} 1/(1-x^j)^binomial(j+k-1,k-1), k>=1, then log(a(n)) ~ (1+1/k) * k^(1/(k+1)) * Zeta(k+1)^(1/(k+1)) * n^(k/(k+1)).
%H Vaclav Kotesovec, <a href="/A255052/b255052.txt">Table of n, a(n) for n = 0..1000</a>
%H Vaclav Kotesovec, <a href="/A255052/a255052.jpg">Graph - The asymptotic ratio</a>
%F G.f.: Product_{j>=1} 1/(1-x^j)^C(j+4,4).
%F a(n) ~ Pi^(49/288) * exp(25/144 - 105*Zeta(3) / (8*Pi^2) + 5*Zeta'(-3)/12 + 29299*Zeta(5) / (128*Pi^4) + 2480625 * Zeta(3) * Zeta(5)^2 / (2*Pi^12) - 72930375 * Zeta(5)^3 / (2*Pi^14) + 1063324867500 * Zeta(5)^5 / Pi^24 + 41 * 7^(1/6) * Pi * n^(1/6) / (768*3^(1/2)) - 2625 * 3^(1/2) * 7^(1/6) * Zeta(3) * Zeta(5) * n^(1/6) / (2*Pi^7) + 540225 * 3^(1/2) * 7^(1/6) * Zeta(5)^2 * n^(1/6) / (16*Pi^9) - 4740474375 * 3^(1/2) * 7^(1/6) * Zeta(5)^4 * n^(1/6) / (4*Pi^19) + 25 * 7^(1/3) * Zeta(3) * n^(1/3) / (4*Pi^2) - 735 * 7^(1/3) * Zeta(5) * n^(1/3) / (8*Pi^4) + 3969000 * 7^(1/3) * Zeta(5)^3 * n^(1/3) / Pi^14 + 7^(3/2) * Pi * n^(1/2) / (3^(3/2)*8) - 4725 * 21^(1/2) * Zeta(5)^2 * n^(1/2) / Pi^9 + 45 * 7^(2/3) * Zeta(5) * n^(2/3) / (2*Pi^4) + 2 * 3^(1/2) * Pi * n^(5/6) / (5 * 7^(1/6))) / (A^(25/12) * 2^(3/2) * 3^(625/576) * 7^(337/1728) * n^(1201/1728)), where A = A074962 = 1.2824271291... is the Glaisher-Kinkelin constant, Zeta(3) = A002117 = 1.202056903..., Zeta(5) = A013663 = 1.036927755... and Zeta'(-3) = ((gamma + log(2*Pi) - 11/6)/30 - 3*Zeta'(4)/Pi^4)/4 = 0.0053785763577743... .
%p with(numtheory):
%p a:= proc(n) option remember; local d, j; `if`(n=0, 1,
%p add(add(d*binomial(d+4, 4), d=divisors(j))*a(n-j), j=1..n)/n)
%p end:
%p seq(a(n), n=0..50); # after Alois P. Heinz
%t nmax=50; CoefficientList[Series[Product[1/(1-x^j)^Binomial[j+4,4],{j,1,nmax}],{x,0,nmax}],x]
%Y Column k=5 of A075196.
%K nonn
%O 0,2
%A _Vaclav Kotesovec_, Mar 08 2015
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