%I #31 Nov 27 2021 11:30:24
%S 1,1,4,35,469,8502,194807,5402497,175985390,6587650497,278674144201,
%T 13148017697608,684554867667117,38988819551585477,2411411875573335044,
%U 160951864352781351959,11531509389384310870257
%N a(n) = Sum_{k=0..n} (-1)^(n-k) * (2*k+1)^(k-1) * Stirling2(n,k).
%H Seiichi Manyama, <a href="/A349527/b349527.txt">Table of n, a(n) for n = 0..356</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/LambertW-Function.html">Lambert W-Function</a>.
%F E.g.f. satisfies: log(A(x)) = (1 - exp(-x)) * A(x)^2.
%F E.g.f.: exp( -LambertW(2 * (exp(-x) - 1))/2 ).
%F G.f.: Sum_{k>=0} (2*k+1)^(k-1) * x^k/Product_{j=1..k} (1 + j*x).
%F a(n) ~ sqrt(2*exp(1) - 1) * sqrt(log(2/(2-exp(-1)))) * n^(n-1) / (2 * exp(n - 1/2) * (1 + log(2/(2*exp(1) - 1)))^n). - _Vaclav Kotesovec_, Nov 21 2021
%t a[n_] := Sum[(-1)^(n - k) * (2*k + 1)^(k - 1) * StirlingS2[n, k], {k, 0, n}]; Array[a, 17, 0] (* _Amiram Eldar_, Nov 27 2021 *)
%o (PARI) a(n) = sum(k=0, n, (-1)^(n-k)*(2*k+1)^(k-1)*stirling(n, k, 2));
%o (PARI) my(N=20, x='x+O('x^N)); Vec(serlaplace(exp(-lambertw(2*(exp(-x)-1))/2)))
%o (PARI) my(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)))
%Y Cf. A008277, A058864, A196555, A349524, A349528.
%K nonn
%O 0,3
%A _Seiichi Manyama_, Nov 20 2021
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