This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A385751 #7 Jul 11 2025 14:51:53
%S A385751 1,1,5,100,5137,539851,101035441,30669875230,14117057058945,
%T A385751 9364637252286181,8603755430968248301,10603853731438585516856,
%U A385751 17077610933602804111318705,35160631271792580418277658415,90839446923946068488317221868825,289828370988497912073923950177143826,1126236403418687405801564385561640043521
%N A385751 a(n) = Sum_{k=0..n} |Stirling1(n,k)| * (n!/k!)^2.
%F A385751 Sum_{n>=0} a(n) * x^n / n!^3 = Sum_{k>=0} (-log(1 - x))^k / k!^3.
%F A385751 a(n) ~ (sqrt(2*Pi/3) * exp(3*log(n)^(1/3) - 3*n) * n^(3*n + 1/2) / log(n)) * (1 - 2/(9*log(n)^(1/3)) + (gamma - 4/81)/log(n)^(2/3) - (40/2187 + 11*gamma/9)/log(n)), where gamma is the Euler-Mascheroni constant A001620. - _Vaclav Kotesovec_, Jul 09 2025
%t A385751 Table[Sum[Abs[StirlingS1[n, k]] (n!/k!)^2, {k, 0, n}], {n, 0, 16}]
%t A385751 nmax = 16; CoefficientList[Series[Sum[(-Log[1 - x])^k/k!^3, {k, 0, nmax}], {x, 0, nmax}], x] Range[0, nmax]!^3
%Y A385751 Cf. A119390, A320502, A385750, A385752.
%K A385751 nonn
%O A385751 0,3
%A A385751 _Ilya Gutkovskiy_, Jul 08 2025