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A323619 Expansion of e.g.f. 1 - LambertW(-log(1+x))*(2 + LambertW(-log(1+x)))/2.

%I A323619 #17 Sep 08 2022 08:46:23
%S A323619 1,1,0,2,3,44,260,3534,40796,658440,11066184,220005840,4750650432,
%T A323619 114430365048,2993377996440,85208541290040,2611784941760640,
%U A323619 85941161628865344,3018822193183216320,112805065528683216192,4467115744449046110720,186900232401341222964480,8237944325702047624948224
%N A323619 Expansion of e.g.f. 1 - LambertW(-log(1+x))*(2 + LambertW(-log(1+x)))/2.
%H A323619 Robert Israel, <a href="/A323619/b323619.txt">Table of n, a(n) for n = 0..398</a>
%F A323619 a(n) = Sum_{k=0..n} Stirling1(n,k)*A000272(k).
%F A323619 a(n) ~ n^(n-2) / ((exp(exp(-1))-1)^(n - 3/2) * exp(n - 3*(1 - exp(-1))/2)). - _Vaclav Kotesovec_, Jan 20 2019
%p A323619 seq(n!*coeff(series(1-LambertW(-log(1+x))*(2+LambertW(-log(1+x)))/2, x=0,23),x,n),n=0..22); # _Paolo P. Lava_, Jan 28 2019
%t A323619 nmax = 22; CoefficientList[Series[1 - LambertW[-Log[1 + x]] (2 + LambertW[-Log[1 + x]])/2, {x, 0, nmax}], x] Range[0, nmax]!
%t A323619 Join[{1}, Table[Sum[StirlingS1[n, k] k^(k - 2), {k, n}], {n, 22}]]
%o A323619 (PARI) {a(n) = if(n==0,1, sum(k=1,n, stirling(n,k,1)*k^(k-2)))};
%o A323619 vector(25, n, n--; a(n)) \\ _G. C. Greubel_, Feb 07 2019
%o A323619 (Magma) [n le 0 select 1 else (&+[StirlingFirst(n,k)*k^(k-2): k in [1..n]]): n in [0..25]]; // _G. C. Greubel_, Feb 07 2019
%o A323619 (Sage) [1] + [sum((-1)^(k+n)*stirling_number1(n,k)*k^(k-2) for k in (1..n)) for n in (1..25)] # _G. C. Greubel_, Feb 07 2019
%Y A323619 Cf. A000272, A038052, A048994, A277489, A305819.
%K A323619 nonn
%O A323619 0,4
%A A323619 _Ilya Gutkovskiy_, Jan 20 2019