A023876 - OEIS

%I #44 Sep 08 2022 08:44:48

%S 1,1,129,2316,26956,385017,5512443,70223666,866470849,10628564312,

%T 126832407040,1469751196093,16694372607012,186350644088784,

%U 2042610304126944,22007441766651756,233482509248479425,2441727926157182541,25187101530316996950,256456174925807404269

%N G.f.: Product_{k>=1} (1 - x^k)^(-k^7).

%H Alois P. Heinz, <a href="/A023876/b023876.txt">Table of n, a(n) for n = 0..1000</a>

%H G. Almkvist, <a href="https://projecteuclid.org/euclid.em/1047674152">Asymptotic formulas and generalized Dedekind sums</a>, Exper. Math., 7 (No. 4, 1998), pp. 343-359.

%H Vaclav Kotesovec, <a href="/A023876/a023876.jpg">Graph - The asymptotic ratio for 10000 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. 21.

%F a(n) ~ (35*Zeta(9))^(119/2160) * exp((3/2)^(20/9) * n^(8/9) * (35*Zeta(9))^(1/9) + Zeta'(-7)) / (2^(247/2160) * 3^(961/1080) * sqrt(Pi) * n^(1199/2160)), where Zeta(9) = A013667 = 1.0020083928260822144..., Zeta'(-7) = ((gamma + log(2*Pi) - 363/140)/30 - 315*Zeta'(8)/Pi^8)/8 = -0.00072864268015924... . - _Vaclav Kotesovec_, Feb 27 2015

%p with(numtheory):

%p a:= proc(n) option remember; `if`(n=0, 1,

%p       add(add(d*d^7, d=divisors(j)) *a(n-j), j=1..n)/n)

%p     end:

%p seq(a(n), n=0..20);  # _Alois P. Heinz_, Nov 02 2012

%t max = 19; Series[ Product[1/(1 - x^k)^k^7, {k, 1, max}], {x, 0, max}] // CoefficientList[#, x] & (* _Jean-François Alcover_, Mar 05 2013 *)

%o (PARI) m=20; x='x+O('x^m); Vec(prod(k=1, m, 1/(1-x^k)^k^7)) \\ _G. C. Greubel_, Oct 31 2018

%o (Magma) m:=20; R<x>:=PowerSeriesRing(Rationals(), m); Coefficients(R! ( (&*[1/(1-x^k)^k^7: k in [1..m]]) )); // _G. C. Greubel_, Oct 31 2018

%Y Column k=7 of A144048.

%K nonn

%O 0,3

%A _Olivier Gérard_

%E Definition corrected by _Franklin T. Adams-Watters_ and _R. J. Mathar_, Dec 04 2006