|
%I #25 Mar 09 2015 09:50:05
%S 1,4,20,80,305,1072,3622,11676,36450,110240,324936,935076,2635338,
%T 7285560,19795370,52930360,139462956,362471020,930186694,2358867240,
%U 5915606398,14680528648,36073675792,87816701332,211891552280,506981067168,1203337174120,2834401172172
%N G.f.: Product_{j>=1} 1/(1-x^j)^binomial(j+3,3).
%H Vaclav Kotesovec, <a href="/A255050/b255050.txt">Table of n, a(n) for n = 0..1000</a>
%H Vaclav Kotesovec, <a href="/A255050/a255050.jpg">Graph - The asymptotic ratio</a>
%F G.f.: Product_{j>=1} 1/(1-x^j)^C(j+3,3).
%F a(n) ~ Zeta(5)^(829/3600) * exp(11/72 - Zeta(3)/(4*Pi^2) + Zeta'(-3)/6 - 121*Zeta(3)^2 / (360*Zeta(5)) - Pi^6/(1800*Zeta(5)) + 11*Pi^8*Zeta(3) / (108000*Zeta(5)^2) - Pi^16/(194400000*Zeta(5)^3) + Pi^2 * n^(1/5)/ (6*2^(2/5) * Zeta(5)^(1/5)) - 11*Pi^4 * Zeta(3) * n^(1/5) / (900*2^(2/5)*Zeta(5)^(6/5)) + Pi^12 * n^(1/5) / (1350000 * 2^(2/5) * Zeta(5)^(11/5)) + 11*Zeta(3) * n^(2/5) / (6*2^(4/5) * Zeta(5)^(2/5)) - Pi^8 * n^(2/5) / (9000 * 2^(4/5) * Zeta(5)^(7/5)) + Pi^4 * n^(3/5) / (90 * 2^(1/5) * Zeta(5)^(3/5)) + 5 * Zeta(5)^(1/5) * n^(4/5) / 2^(8/5)) / (A^(11/6) * 2^(971/1800) * 5^(1/2) * Pi * n^(2629/3600)), 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+3, 3), 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+3,3],{j,1,nmax}],{x,0,nmax}],x]
%Y Column k=4 of A075196.
%Y Cf. A005380, A217093, A255052.
%K nonn
%O 0,2
%A _Vaclav Kotesovec_, Mar 08 2015
|
