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 A285340 Expansion of Product_{k>=0} (1 + x^(5*k+4))^(5*k+4). 3

%I #19 Apr 20 2017 03:15:16

%S 1,0,0,0,4,0,0,0,6,9,0,0,4,36,14,0,1,54,92,19,0,36,228,202,24,9,272,

%T 702,358,29,158,1168,1696,598,70,1027,3810,3605,904,501,4600,10196,

%U 6898,1408,3078,15805,24104,12242,2838,14103,46090,51376,20566,9443,51682

%N Expansion of Product_{k>=0} (1 + x^(5*k+4))^(5*k+4).

%C In general, if m > 1 and g.f. = Product_{k>=1} (1 + x^(m*k-1))^(m*k-1), then a(n, m) ~ exp(2^(-4/3) * 3^(4/3) * m^(-1/3) * Zeta(3)^(1/3) * n^(2/3)) * Zeta(3)^(1/6) / (2^(1/6 + 1/(2*m) + m/12) * 3^(1/3) * m^(1/6) * sqrt(Pi) * n^(2/3)). - _Vaclav Kotesovec_, Apr 17 2017

%H Seiichi Manyama, <a href="/A285340/b285340.txt">Table of n, a(n) for n = 0..10000</a>

%F a(n) ~ exp(2^(-4/3) * 3^(4/3) * 5^(-1/3) * Zeta(3)^(1/3) * n^(2/3)) * Zeta(3)^(1/6) / (2^(41/60) * 3^(1/3) * 5^(1/6) * sqrt(Pi) * n^(2/3)). - _Vaclav Kotesovec_, Apr 17 2017

%t nmax = 50; CoefficientList[Series[Product[(1 + x^(5*k-1))^(5*k-1), {k,1,nmax}], {x,0,nmax}], x] (* _Vaclav Kotesovec_, Apr 17 2017 *)

%Y Product_{k>=0} (1 + x^(m*k+m-1))^(m*k+m-1): A262736 (m=2), A262948 (m=3), A285339 (m=4), this sequence (m=5).

%Y Cf. A285214, A285338.

%K nonn

%O 0,5

%A _Seiichi Manyama_, Apr 17 2017

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Last modified August 7 17:47 EDT 2024. Contains 375017 sequences. (Running on oeis4.)