|
%I #52 Aug 09 2021 11:19:54
%S 1,6,25,100,511,3626,30045,262800,2470171,25889446,302003065,
%T 3821936300,51672723831,745789322466,11505096936085,189023074558600,
%U 3288243760145491,60319276499454686,1164282909466221105,23603464830964817700,501435697062735519151
%N E.g.f.: -log( 1 - (exp(x) - 1)^3 / 3! ).
%F a(n) = Stirling2(n,3) + (1/n) * Sum_{k=1..n-1} binomial(n,k) * Stirling2(n-k,3) * k * a(k).
%F a(n) ~ (n-1)! / (log(6^(1/3)+1))^n. - _Vaclav Kotesovec_, Aug 09 2021
%t nmax = 23; CoefficientList[Series[-Log[1 - (Exp[x] - 1)^3/3!], {x, 0, nmax}], x] Range[0, nmax]! // Drop[#, 3] &
%t a[n_] := a[n] = StirlingS2[n, 3] + (1/n) Sum[Binomial[n, k] StirlingS2[n - k, 3] k a[k], {k, 1, n - 1}]; Table[a[n], {n, 3, 23}]
%o (PARI) my(x='x+O('x^25)); Vec(serlaplace(-log(1-(exp(x)-1)^3/3!))) \\ _Michel Marcus_, Aug 09 2021
%Y Cf. A000392, A000629, A003704, A327504, A346894, A346954, A346955.
%K nonn
%O 3,2
%A _Ilya Gutkovskiy_, Aug 08 2021
|
