A357026 - OEIS

%I #19 Feb 16 2025 08:34:04

%S 1,0,2,6,58,460,5528,70308,1098060,18910512,371480832,8022952080,

%T 191325228576,4961955705408,139572074260656,4224646630879920,

%U 137077496211066384,4744151145076980864,174517898073769832448,6798949897214608689024,279688643858492900930496

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

%H Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/LambertW-Function.html">Lambert W-Function</a>.

%F E.g.f. satisfies log(A(x)) = log(1 - x)^2 * A(x).

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

%F E.g.f.: A(x) = Sum_{k>=0} (k+1)^(k-1) * log(1 - x)^(2*k) / k!.

%F E.g.f.: A(x) = exp( -LambertW(-log(1-x)^2) ).

%F E.g.f.: A(x) = -LambertW(-log(1 - x)^2)/log(1 - x)^2.

%t m = 21; (* number of terms *)

%t A[_] = 0;

%t Do[A[x_] = (1 - x)^(Log[1 - x]*A[x]) + O[x]^m // Normal, {m}];

%t CoefficientList[A[x], x]*Range[0, m - 1]! (* _Jean-François Alcover_, Sep 12 2022 *)

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

%o (PARI) my(N=30, x='x+O('x^N)); Vec(serlaplace(sum(k=0, N, (k+1)^(k-1)*log(1-x)^(2*k)/k!)))

%o (PARI) my(N=30, x='x+O('x^N)); Vec(serlaplace(exp(-lambertw(-log(1-x)^2))))

%o (PARI) my(N=30, x='x+O('x^N)); Vec(serlaplace(-lambertw(-log(1-x)^2)/log(1-x)^2))

%Y Cf. A052813, A357027.

%K nonn,changed

%O 0,3

%A _Seiichi Manyama_, Sep 09 2022