A355842 - OEIS

%I #27 Aug 28 2022 04:24:22

%S 1,0,2,3,44,210,3054,27300,449952,6020784,115381080,2053568880,

%T 45733246536,1010390340960,25916586868704,680621684914080,

%U 19881379012231680,603034125051738240,19833651290982164544,680927283288289169280,24953207662252739030400

%N E.g.f. satisfies A(x) = 1/(1 - x)^(x * A(x)).

%H Vaclav Kotesovec, <a href="/A355842/b355842.txt">Table of n, a(n) for n = 0..400</a>

%F E.g.f.: exp( -LambertW(x * log(1-x)) ).

%F E.g.f.: LambertW(x * log(1-x)) / (x * log(1-x)).

%F a(n) ~ sqrt(1 + exp(1)*r^2/(1-r)) * n^(n-1) / (exp(n-1) * r^n), where r = 0.5123112855238643734867005914814802444318611742265... is the positive root of the equation r*log(1-r) = -exp(-1). - _Vaclav Kotesovec_, Jul 21 2022

%F a(n) = n! * Sum_{k=0..floor(n/2)} (k+1)^(k-1) * |Stirling1(n-k,k)|/(n-k)!. - _Seiichi Manyama_, Aug 28 2022

%t nmax = 20; CoefficientList[Series[LambertW[x * Log[1-x]] / (x * Log[1-x]), {x, 0, nmax}], x] * Range[0, nmax]! (* _Vaclav Kotesovec_, Jul 21 2022 *)

%o (PARI) my(N=20, x='x+O('x^N)); Vec(serlaplace(exp(-lambertw(x*log(1-x)))))

%o (PARI) a(n) = n!*sum(k=0, n\2, (k+1)^(k-1)*abs(stirling(n-k, k, 1))/(n-k)!); \\ _Seiichi Manyama_, Aug 28 2022

%Y Cf. A052813, A184949, A355779.

%K nonn

%O 0,3

%A _Seiichi Manyama_, Jul 18 2022