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A023873 Expansion of Product_{k>=1} (1 - x^k)^(-k^4).

%I A023873 #61 Jan 29 2025 22:30:03
%S A023873 1,1,17,98,490,2411,11940,56093,256274,1140980,4977222,21273772,
%T A023873 89281011,368408970,1496993290,5996312751,23700208290,92510062036,
%U A023873 356887002352,1361671469470,5141380256124,19221678032134,71190778935805,261320839754142,951091521384860
%N A023873 Expansion of Product_{k>=1} (1 - x^k)^(-k^4).
%C A023873 Let A(x) denote the g.f. and let m be an integer. Define a sequence by u(n) = [x^n] A(x)^(m*n). We conjecture that the supercongruence u(n*p^r) == u(n*p^(r-1)) (mod p^(3*r)) holds for all positive integers n and r and all primes p >= 7. Cf. A380581. - _Peter Bala_, Jan 21 2025
%H A023873 Seiichi Manyama, <a href="/A023873/b023873.txt">Table of n, a(n) for n = 0..3339</a> (first 801 terms from Alois P. Heinz)
%H A023873 G. Almkvist, <a href="http://projecteuclid.org/euclid.em/1047674152">Asymptotic formulas and generalized Dedekind sums</a>, Exper. Math., 7 (No. 4, 1998), pp. 343-359.
%H A023873 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 A023873 a(n) ~ exp(Pi * 2^(3/2) * 3^(2/3) * n^(5/6) / (5 * 7^(1/6)) + 3*Zeta(5) / (4*Pi^4)) / (2^(3/4) * 3^(2/3) * 7^(1/12) * n^(7/12)), where Zeta(5) = A013663 = 1.036927755143369926... . - _Vaclav Kotesovec_, Feb 27 2015
%F A023873 G.f.: exp( Sum_{n>=1} sigma_5(n)*x^n/n ). - _Seiichi Manyama_, Mar 04 2017
%F A023873 a(n) = (1/n)*Sum_{k=1..n} sigma_5(k)*a(n-k). - _Seiichi Manyama_, Mar 04 2017
%p A023873 with(numtheory):
%p A023873 a:= proc(n) option remember; `if`(n=0, 1,
%p A023873       add(add(d*d^4, d=divisors(j)) *a(n-j), j=1..n)/n)
%p A023873     end:
%p A023873 seq(a(n), n=0..25);  # _Alois P. Heinz_, Nov 02 2012
%t A023873 max = 27; Series[ Product[1/(1 - x^k)^k^4, {k, 1, max}], {x, 0, max}] // CoefficientList[#, x] & (* _Jean-François Alcover_, Mar 05 2013 *)
%o A023873 (PARI) m=30; x='x+O('x^m); Vec(prod(k=1, m, 1/(1-x^k)^k^4)) \\ _G. C. Greubel_, Oct 30 2018
%o A023873 (Magma) m:=30; R<x>:=PowerSeriesRing(Rationals(), m); Coefficients(R! ( (&*[1/(1-x^k)^k^4: k in [1..m]]) )); // _G. C. Greubel_, Oct 30 2018
%Y A023873 Column k=4 of A144048.
%Y A023873 Cf. A283271, A380581.
%K A023873 nonn,easy
%O A023873 0,3
%A A023873 _Olivier Gérard_
%E A023873 Definition corrected by _Franklin T. Adams-Watters_ and _R. J. Mathar_, Dec 04 2006