cp's OEIS Frontend

A192065 Expansion of Product_{k>=1} Q(x^k)^k where Q(x) = Product_{k>=1} (1 + x^k).

%I A192065 #46 Sep 30 2024 19:09:04
%S A192065 1,1,3,7,14,28,58,106,201,372,669,1187,2101,3624,6229,10591,17796,
%T A192065 29659,49107,80492,131157,212237,341084,544883,865717,1367233,2148552,
%U A192065 3359490,5227270,8096544,12486800,19174319,29326306,44678825,67811375,102549673,154545549
%N A192065 Expansion of Product_{k>=1} Q(x^k)^k where Q(x) = Product_{k>=1} (1 + x^k).
%C A192065 Euler transform of A002131. - _Vaclav Kotesovec_, Mar 26 2018
%H A192065 Seiichi Manyama, <a href="/A192065/b192065.txt">Table of n, a(n) for n = 0..10000</a> (terms 0..1000 from T. D. Noe)
%H A192065 Lida Ahmadi, Ricardo Gómez Aíza, and Mark Daniel Ward, <a href="https://doi.org/10.1007/s44007-024-00134-w">A unified treatment of families of partition functions</a>, La Matematica (2024). Preprint available as <a href="https://arxiv.org/abs/2303.02240">arXiv:2303.02240</a> [math.CO], 2023.
%F A192065 a(0) = 1, a(n) = (1/n)*Sum_{k=1..n} A288418(k)*a(n-k) for n > 0. - _Seiichi Manyama_, Jun 09 2017
%F A192065 a(n) ~ exp(3*Pi^(2/3) * Zeta(3)^(1/3) * n^(2/3)/2^(5/3) - Pi^(4/3) * n^(1/3) / (3*2^(7/3) * Zeta(3)^(1/3)) - Pi^2 / (864 * Zeta(3))) * Zeta(3)^(1/6) / (2^(19/24) * sqrt(3) * Pi^(1/6) * n^(2/3)). - _Vaclav Kotesovec_, Mar 23 2018
%t A192065 nn = 30; b = Table[DivisorSigma[1, n], {n, nn}]; CoefficientList[Series[Product[(1 + x^m)^b[[m]], {m, nn}], {x, 0, nn}], x] (* _T. D. Noe_, Jun 19 2012 *)
%t A192065 kmax = 37; Product[QPochhammer[-1, x^k]^k/2^k, {k, 1, kmax}] + O[x]^kmax // CoefficientList[#, x]& (* _Jean-François Alcover_, Jul 03 2017 *)
%t A192065 nmax = 40; CoefficientList[Series[Exp[Sum[Sum[DivisorSum[k, # / GCD[#, 2] &] * x^(j*k) / j, {k, 1, Floor[nmax/j] + 1}], {j, 1, nmax}]], {x, 0, nmax}], x] (* _Vaclav Kotesovec_, Mar 31 2018 *)
%o A192065 (PARI) N=66;  x='x+O('x^N);
%o A192065 Q(x)=prod(k=1,N,1+x^k);
%o A192065 gf=prod(k=1,N, Q(x^k)^k );
%o A192065 Vec(gf) /* _Joerg Arndt_, Jun 24 2011 */
%Y A192065 Cf. A061256 (1/Product_{k>=1} P(x^k)^k where P(x) = Product_{k>=1} (1 - x^k)).
%Y A192065 Product_{k>=1} (1 + x^k)^sigma_m(k): A107742 (m=0), this sequence (m=1), A288414 (m=2), A288415 (m=3), A301548 (m=4), A301549 (m=5), A301550 (m=6), A301551 (m=7), A301552 (m=8).
%K A192065 nonn
%O A192065 0,3
%A A192065 _Joerg Arndt_, Jun 24 2011