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A255528 G.f.: Product_{k>=1} 1/(1+x^k)^k. 28

%I #37 Apr 13 2017 17:57:15

%S 1,-1,-1,-2,1,0,4,2,8,-2,4,-11,-1,-25,-5,-35,13,-26,49,-6,110,6,159,

%T -23,182,-141,129,-358,62,-640,39,-897,237,-1013,771,-914,1793,-664,

%U 3143,-565,4635,-1157,5727,-3119,6121,-7041,5642,-13088,5097,-20758,5879

%N G.f.: Product_{k>=1} 1/(1+x^k)^k.

%C In general, if m >= 1 and g.f. = Product_{k>=1} 1/(1 + x^k)^(m*k), then a(n, m) ~ (-1)^n * exp(-m/12 + 3 * 2^(-5/3) * m^(1/3) * Zeta(3)^(1/3) * n^(2/3)) * 2^(m/18 - 5/6) * A^m * m^(1/6 - m/36) * Zeta(3)^(1/6 - m/36) * n^(m/36 - 2/3) / sqrt(3*Pi), where A is the Glaisher-Kinkelin constant A074962. - _Vaclav Kotesovec_, Apr 13 2017

%H Seiichi Manyama, <a href="/A255528/b255528.txt">Table of n, a(n) for n = 0..10000</a> (terms 0..5000 from Vaclav Kotesovec)

%H Vaclav Kotesovec, <a href="/A255528/a255528_1.jpg">Graph - the asymptotic ratio (250000 terms)</a>

%H Vaclav Kotesovec, <a href="http://arxiv.org/abs/1509.08708">A method of finding the asymptotics of q-series based on the convolution of generating functions</a>, arXiv:1509.08708 [math.CO], Sep 30 2015, p. 20.

%F a(n) ~ (-1)^n * A * Zeta(3)^(5/36) * exp(3*Zeta(3)^(1/3)*n^(2/3)/2^(5/3) - 1/12) / (2^(7/9) * sqrt(3*Pi) * n^(23/36)), where Zeta(3) = A002117 and A = A074962 is the Glaisher-Kinkelin constant. - _Vaclav Kotesovec_, Sep 29 2015

%F a(0) = 1, a(n) = -(1/n)*Sum_{k=1..n} A078306(k)*a(n-k) for n > 0. - _Seiichi Manyama_, Apr 06 2017

%p with(numtheory): A000219:=proc(n) option remember; if n = 0 then 1 else add(sigma[2](k)*A000219(n-k), k = 1..n)/n fi: end: A073592:=proc(n) option remember; if n = 0 then 1 else -add(sigma[2](k)*A073592(n-k), k = 1..n)/n fi: end: a:=proc(n); add(A073592(n-2*m)*A000219(m), m = 0..floor(n/2)): end: seq(a(n), n = 0..50); # _Vaclav Kotesovec_, Mar 09 2015

%t nmax=100; CoefficientList[Series[Product[1/(1+x^k)^k,{k,1,nmax}],{x,0,nmax}],x]

%o (PARI) {a(n) = if(n<0, 0, polcoeff(exp(sum(k=1, n, (-1)^k * x^k / (1-x^k)^2 / k, x*O(x^n))), n))}

%o for(n=0, 100, print1(a(n), ", "))

%Y Cf. A000219, A026007, A026011, A073592, A262736.

%Y Cf. A278710 (m=2), A279031 (m=3), A279411 (m=4), A279932 (m=5).

%K sign

%O 0,4

%A _Vaclav Kotesovec_, Feb 24 2015

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