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G.f.: exp( Sum_{n>=1} A174468(n)*x^n/n ) where A174468(n) = Sum_{d|n} d*sigma(n/d)*sigma(d).
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%I #13 Sep 30 2024 21:16:48

%S 1,1,5,10,31,58,157,299,711,1367,2987,5679,11807,22117,44006,81513,

%T 156885,286413,537058,967367,1773882,3155223,5677183,9976095,17661695,

%U 30682683,53544796,92037152,158575796,269850363,459636546,774851829

%N G.f.: exp( Sum_{n>=1} A174468(n)*x^n/n ) where A174468(n) = Sum_{d|n} d*sigma(n/d)*sigma(d).

%H 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 From _Ricardo Gómez Aíza_, Mar 08 2023: (Start)

%F E.g.f.: Product_{n>=1,m>=1,k>=1} 1 / (1 - x^(n * m * k))^n.

%F log(a(n) / n!) ~ (3/2) * (Zeta(3) * Pi^4 / 18)^(1/3) * n^(2/3). (End)

%o (PARI) {a(n)=polcoeff(exp(sum(m=1,n,x^m/m*sumdiv(m,d,d*sigma(m/d)*sigma(d)))+x*O(x^n)),n)}

%Y Cf. A174465, A000203 (sigma).

%K nonn

%O 0,3

%A _Paul D. Hanna_, Apr 04 2010