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A023871 Expansion of Product_{k>=1} (1 - x^k)^(-k^2).

%I A023871 #68 Feb 03 2025 21:28:03
%S A023871 1,1,5,14,40,101,266,649,1593,3765,8813,20168,45649,101591,223654,
%T A023871 486046,1045541,2225167,4692421,9804734,20318249,41766843,85218989,
%U A023871 172628766,347338117,694330731,1379437080,2724353422,5350185097,10449901555,20304465729,39254599832
%N A023871 Expansion of Product_{k>=1} (1 - x^k)^(-k^2).
%C A023871 In general, if g.f. = Product_{k>=1} 1/(1 - x^k)^(c2*k^2 + c1*k + c0) and c2 > 0, then a(n) ~ exp(4*Pi * c2^(1/4) * n^(3/4) / (3*15^(1/4)) + c1*Zeta(3) / Pi^2 * sqrt(15*n/c2) + (Pi * 5^(1/4) * c0 / (2*3^(3/4) * c2^(1/4)) - 15^(5/4) * c1^2 * Zeta(3)^2 / (2*c2^(5/4) * Pi^5)) * n^(1/4) + c1/12 + 75 * c1^3 * Zeta(3)^3 / (c2^2 * Pi^8) - 5*c0 * c1 * Zeta(3) / (4*c2 * Pi^2) - c2*Zeta(3) / (4*Pi^2)) * Pi^(c1/12) * (c2/15)^(1/8 + c0/8 + c1/48) / (A^c1 * 2^((c0 + 3)/2) * n^(5/8 + c0/8 + c1/48)), where A is the Glaisher-Kinkelin constant A074962. - _Vaclav Kotesovec_, Nov 08 2017
%C A023871 Let A(x) = Product_{k >= 1} (1 - x^k)^(-k^2). The sequence defined by u(n) := [x^n] A(x)^n is conjectured to satisfy the supercongruences u(n*p^r) == u(n*p^(r-1)) (mod p^(3*r)) for all primes p >= 7 and all positive integers n and r. See A380290. - _Peter Bala_, Feb 02 2025
%C A023871 a(n) is the number of partitions of n where there are k^2 sorts of part k. - _Joerg Arndt_, Feb 02 2025
%H A023871 Seiichi Manyama, <a href="/A023871/b023871.txt">Table of n, a(n) for n = 0..10000</a> (first 1001 terms from Alois P. Heinz)
%H A023871 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 A023871 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 A023871 a(n) = 1/n * Sum_{k=1..n} a(n-k)*sigma_3(k), n > 0, a(0)=1, where sigma_3(n) = A001158(n) = sum of cubes of divisors of n. - _Vladeta Jovovic_, Jan 20 2002
%F A023871 G.f.: Prod_{n>=1} exp(sigma_3(n)*x^n/n), where sigma_3(n) is the sum of cubes of divisors of n (=A001158(n)). - _N-E. Fahssi_, Mar 28 2010
%F A023871 G.f. (conjectured): 1/Product_{n>=1} E(x^n)^J2(n) where E(x) = Product_{n>=1} 1-x^n and J2(n) = A007434(n) [follows from the identity Sum_{d|n} J2(d) = n^2 - _Peter Bala_, Feb 02 2025]. - _Joerg Arndt_, Jan 25 2011
%F A023871 a(n) ~ exp(4 * Pi * n^(3/4) / (3^(5/4) * 5^(1/4)) - Zeta(3) / (4*Pi^2)) / (2^(3/2) * 15^(1/8) * n^(5/8)), where Zeta(3) = A002117 = 1.2020569031595942853997... . - _Vaclav Kotesovec_, Feb 27 2015
%p A023871 with(numtheory):
%p A023871 a:= proc(n) option remember; `if`(n=0, 1,
%p A023871       add(add(d*d^2, d=divisors(j)) *a(n-j), j=1..n)/n)
%p A023871     end:
%p A023871 seq(a(n), n=0..35); # _Alois P. Heinz_, Nov 02 2012
%t A023871 max = 31; Series[ Product[ 1/(1-x^k)^k^2, {k, 1, max}], {x, 0, max}] // CoefficientList[#, x]& (* _Jean-François Alcover_, Mar 05 2013 *)
%o A023871 (PARI) m=40; x='x+O('x^m); Vec(prod(k=1, m, 1/(1-x^k)^k^2)) \\ _G. C. Greubel_, Oct 29 2018
%o A023871 (Magma) m:=40; R<x>:=PowerSeriesRing(Rationals(), m); Coefficients(R! ( (&*[1/(1-x^k)^k^2: k in [1..m]]) )); // _G. C. Greubel_, Oct 29 2018
%o A023871 (SageMath) # uses[EulerTransform from A166861]
%o A023871 b = EulerTransform(lambda n: n^2)
%o A023871 print([b(n) for n in range(32)]) # _Peter Luschny_, Nov 11 2020
%Y A023871 Euler transform of squares (A000290).
%Y A023871 Cf. A000219, A023872-A023878, A294530, A380290.
%Y A023871 Column k=2 of A144048. - _Alois P. Heinz_, Nov 02 2012
%K A023871 nonn,easy
%O A023871 0,3
%A A023871 _Olivier Gérard_
%E A023871 Definition corrected by _Franklin T. Adams-Watters_ and _R. J. Mathar_, Dec 04 2006