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%I #20 Dec 31 2023 14:29:07
%S 1,2,14,172,2964,65848,1789688,57521280,2133964352,89744964288,
%T 4219022123328,219246630903936,12479659844383104,772174659456713472,
%U 51603153976362554112,3704166182571098222592,284239227254465994240000,23218955083323248158556160
%N a(0) = 1; thereafter a(n) = Sum_{k=1..n} (2*k)^k * Stirling1(n,k).
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/LambertW-Function.html">Lambert W-Function</a>.
%F E.g.f.: 1/(1 + LambertW( -2 * log(1+x) )), where LambertW() is the Lambert W-function.
%F a(n) ~ n^n / (sqrt(2) * (exp(exp(-1)/2) - 1)^(n+1/2) * exp(n - exp(-1)/4 + 1/2)). - _Vaclav Kotesovec_, Feb 06 2022
%t Join[{1},Table[Sum[(2k)^k StirlingS1[n,k],{k,n}],{n,20}]] (* _Harvey P. Dale_, Dec 31 2023 *)
%o (PARI) a(n) = sum(k=0, n, (2*k)^k*stirling(n, k, 1));
%o (PARI) my(N=20, x='x+O('x^N)); Vec(serlaplace(1/(1+lambertw(-2*log(1+x)))))
%Y Cf. A305819, A351182, A351275, A351276.
%K nonn
%O 0,2
%A _Seiichi Manyama_, Feb 05 2022
%E Made a(0) = 1 explicit and changed range of k in definition to start at 1 at the suggestion of _Harvey P. Dale_. - _N. J. A. Sloane_, Dec 31 2023
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