%I #32 Jul 27 2023 17:41:19
%S 1,1,5,49,856,22376,842536,42409480,2782192064,229357803456,
%T 23289083584704,2851295406197184,414855423241758720,
%U 70695451937596732416,13958230719814052097024,3159974451734082088897536,813380358295803762813321216,236172126115504055456155975680
%N G.f.: Product_{n>=1} 1/(1 - x^n/n^2) = Sum_{n>=0} a(n)*x^n/n!^2.
%H Paul D. Hanna, <a href="/A249588/b249588.txt">Table of n, a(n) for n = 0..100</a>
%H MathOverflow, <a href="https://mathoverflow.net/questions/404805/asymptotic-growth-of-a-sum-involving-partitions">asymptotic growth of a sum involving partitions</a>, Sep 26 2021.
%F a(n) = Sum_{k=1..n} n!*(n-1)!/(n-k)!^2 * b(k) * a(n-k), where b(k) = Sum_{d|k} d^(1-2*k/d) and a(0) = 1 (after Vladeta Jovovic in A007841).
%F a(n) ~ 2 * n!^2. - _Vaclav Kotesovec_, Mar 05 2016
%e G.f.: A(x) = 1 + x + 5*x^2/2!^2 + 49*x^3/3!^2 + 856*x^4/4!^2 +...
%e where
%e A(x) = 1/((1-x)*(1-x^2/4)*(1-x^3/9)*(1-x^4/16)*(1-x^5/25)*...).
%p b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
%p b(n, i-1)+b(n-i, min(i, n-i))*((i-1)!*binomial(n, i))^2 ))
%p end:
%p a:= n-> b(n$2):
%p seq(a(n), n=0..17); # _Alois P. Heinz_, Jul 27 2023
%t b[k_] := b[k] = DivisorSum[k, #^(1-2*k/#) &]; a[0] = 1; a[n_] := a[n] = Sum[n!*(n-1)!/(n-k)!^2*b[k]*a[n-k], {k, 1, n}]; Table[a[n], {n, 0, 20}] (* _Jean-François Alcover_, Dec 23 2015, adapted from PARI *)
%t Table[n!^2 * SeriesCoefficient[Product[1/(1 - x^m/m^2), {m, 1, n}], {x, 0, n}], {n, 0, 20}] (* _Vaclav Kotesovec_, Mar 05 2016 *)
%o (PARI) {a(n)=n!^2*polcoeff(prod(k=1, n, 1/(1-x^k/k^2 +x*O(x^n))),n)}
%o for(n=0,20,print1(a(n),", "))
%o (PARI) /* Using logarithmic derivative: */
%o {b(k) = sumdiv(k,d, d^(1-2*k/d))}
%o {a(n) = if(n==0,1,sum(k=1,n, n!*(n-1)!/(n-k)!^2 * b(k) * a(n-k)))}
%o for(n=0,20,print1(a(n),", "))
%Y Cf. A249590, A249607, A007841, A249593, A249592, A269791, A269793, A269794.
%K nonn
%O 0,3
%A _Paul D. Hanna_, Nov 01 2014
%E Name clarified by _Vaclav Kotesovec_, Mar 05 2016