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A349524 a(n) = Sum_{k=0..n} (2*k+1)^(k-1) * Stirling2(n,k).

%I A349524 #36 Jul 29 2022 08:42:15
%S A349524 1,1,6,65,1059,23232,642859,21507733,844701160,38108248719,
%T A349524 1942394699283,110401966739110,6923805346540685,474957822716470901,
%U A349524 35377953843680999326,2843665890900123673997,245340865605247369255751,22614510471168438300336440,2217985444621941684970200607
%N A349524 a(n) = Sum_{k=0..n} (2*k+1)^(k-1) * Stirling2(n,k).
%H A349524 Seiichi Manyama, <a href="/A349524/b349524.txt">Table of n, a(n) for n = 0..348</a>
%F A349524 E.g.f.: sqrt(-LambertW(2*(-exp(x) + 1)) / (2*(exp(x) - 1))).
%F A349524 E.g.f.: exp(-LambertW(2 - 2*exp(x))/2).
%F A349524 a(n) ~ c * d^n * n! / n^(3/2), where d = 1/log(1 + 1/(2*exp(1))) and c = sqrt(exp(1) * (1 + 2*exp(1)) * log(1 + 1/(2*exp(1))) / (2*Pi))/2 = 0.3428481589262346912499652905097648170872882109000404115070292580887155335...
%F A349524 a(n) ~ sqrt(1 + 2*exp(1)) * n^(n-1) / (2*exp(n - 1/2) * log(1 + 1/(2*exp(1)))^(n - 1/2)).
%F A349524 E.g.f. satisfies: log(A(x)) = (exp(x) - 1) * A(x)^2.
%F A349524 G.f.: Sum_{k>=0} (2*k+1)^(k-1) * x^k/Product_{j=1..k} (1 - j*x). - _Seiichi Manyama_, Nov 20 2021
%p A349524 b:= proc(n, m) option remember; `if`(n=0,
%p A349524      (2*m+1)^(m-1), m*b(n-1, m)+b(n-1, m+1))
%p A349524     end:
%p A349524 a:= n-> b(n, 0):
%p A349524 seq(a(n), n=0..24);  # _Alois P. Heinz_, Jul 29 2022
%t A349524 Table[Sum[(2*k+1)^(k-1)*StirlingS2[n, k], {k, 0, n}], {n, 0, 20}]
%t A349524 nmax = 20; CoefficientList[Series[Sqrt[-LambertW[2*(-E^x + 1)]/(2*(E^x - 1))], {x, 0, nmax}], x] * Range[0, nmax]!
%o A349524 (PARI) a(n) = sum(k=0, n, (2*k+1)^(k-1)*stirling(n, k, 2)); \\ _Seiichi Manyama_, Nov 20 2021
%o A349524 (PARI) N=20; x='x+O('x^N); Vec(sum(k=0, N, (2*k+1)^(k-1)*x^k/prod(j=1, k, 1-j*x))) \\ _Seiichi Manyama_, Nov 20 2021
%Y A349524 Cf. A000110, A008277, A052880, A349504, A349525.
%K A349524 nonn
%O A349524 0,3
%A A349524 _Vaclav Kotesovec_, Nov 20 2021