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A351218 a(n) = Sum_{k=0..n} (-k)^k * Stirling2(n,k).

%I A351218 #18 Feb 16 2025 08:34:02
%S A351218 1,-1,3,-16,121,-1181,14114,-199543,3257139,-60279214,1247164055,
%T A351218 -28525394481,714681439212,-19465007759913,572609747089735,
%U A351218 -18093710202583480,611202186074834221,-21979340746682042249,838330656532184312218,-33803668628843391999843
%N A351218 a(n) = Sum_{k=0..n} (-k)^k * Stirling2(n,k).
%H A351218 Alois P. Heinz, <a href="/A351218/b351218.txt">Table of n, a(n) for n = 0..400</a>
%H A351218 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/LambertW-Function.html">Lambert W-Function</a>.
%F A351218 E.g.f.: 1/(1 + LambertW(exp(x) - 1)), where LambertW() is the Lambert W-function.
%F A351218 a(n) ~ (-1)^n * n^n / (sqrt(exp(1)-1) * (1 - log(exp(1)-1))^(n + 1/2) * exp(n)). - _Vaclav Kotesovec_, Feb 05 2022
%p A351218 b:= proc(n, m) option remember; `if`(n=0,
%p A351218       (-m)^m, m*b(n-1, m)+b(n-1, m+1))
%p A351218     end:
%p A351218 a:= n-> b(n, 0):
%p A351218 seq(a(n), n=0..20);  # _Alois P. Heinz_, Jul 17 2022
%t A351218 Table[Sum[(-1)^k * k^k * StirlingS2[n,k], {k,1,n}], {n,0,20}] (* _Vaclav Kotesovec_, Feb 05 2022 *)
%o A351218 (PARI) a(n) = sum(k=0, n, (-k)^k*stirling(n, k, 2));
%o A351218 (PARI) my(N=40, x='x+O('x^N)); Vec(serlaplace(1/(1+lambertw(exp(x)-1))))
%Y A351218 Cf. A282190, A305981.
%K A351218 sign
%O A351218 0,3
%A A351218 _Seiichi Manyama_, Feb 05 2022