cp's OEIS Frontend

A357026 E.g.f. satisfies A(x) = (1 - x)^(log(1 - x) * A(x)).

%I A357026 #21 Aug 05 2025 07:06:50
%S A357026 1,0,2,6,58,460,5528,70308,1098060,18910512,371480832,8022952080,
%T A357026 191325228576,4961955705408,139572074260656,4224646630879920,
%U A357026 137077496211066384,4744151145076980864,174517898073769832448,6798949897214608689024,279688643858492900930496
%N A357026 E.g.f. satisfies A(x) = (1 - x)^(log(1 - x) * A(x)).
%H A357026 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/LambertW-Function.html">Lambert W-Function</a>.
%F A357026 E.g.f. satisfies log(A(x)) = log(1 - x)^2 * A(x).
%F A357026 a(n) = Sum_{k=0..floor(n/2)} (2*k)! * (k+1)^(k-1) * |Stirling1(n,2*k)|/k!.
%F A357026 E.g.f.: A(x) = Sum_{k>=0} (k+1)^(k-1) * log(1 - x)^(2*k) / k!.
%F A357026 E.g.f.: A(x) = exp( -LambertW(-log(1-x)^2) ).
%F A357026 E.g.f.: A(x) = -LambertW(-log(1 - x)^2)/log(1 - x)^2.
%F A357026 a(n) ~ sqrt(2) * exp(5/4 + exp(-1/2)/2 - n) * n^(n-1) / (1 - exp(-exp(-1/2)))^(n - 1/2). - _Vaclav Kotesovec_, Aug 05 2025
%t A357026 m = 21; (* number of terms *)
%t A357026 A[_] = 0;
%t A357026 Do[A[x_] = (1 - x)^(Log[1 - x]*A[x]) + O[x]^m // Normal, {m}];
%t A357026 CoefficientList[A[x], x]*Range[0, m - 1]! (* _Jean-François Alcover_, Sep 12 2022 *)
%o A357026 (PARI) a(n) = sum(k=0, n\2, (2*k)!*(k+1)^(k-1)*abs(stirling(n, 2*k, 1))/k!);
%o A357026 (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 A357026 (PARI) my(N=30, x='x+O('x^N)); Vec(serlaplace(exp(-lambertw(-log(1-x)^2))))
%o A357026 (PARI) my(N=30, x='x+O('x^N)); Vec(serlaplace(-lambertw(-log(1-x)^2)/log(1-x)^2))
%Y A357026 Cf. A052813, A357027.
%K A357026 nonn
%O A357026 0,3
%A A357026 _Seiichi Manyama_, Sep 09 2022