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

1, 0, 2, 6, 58, 460, 5528, 70308, 1098060, 18910512, 371480832, 8022952080, 191325228576, 4961955705408, 139572074260656, 4224646630879920, 137077496211066384, 4744151145076980864, 174517898073769832448, 6798949897214608689024, 279688643858492900930496

Offset

0,3

Links

Table of n, a(n) for n=0..20.

Eric Weisstein's World of Mathematics, Lambert W-Function.

Formula

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

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

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

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

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

Mathematica

m = 21; (* number of terms *)
A[_] = 0;
Do[A[x_] = (1 - x)^(Log[1 - x]*A[x]) + O[x]^m // Normal, {m}];
CoefficientList[A[x], x]*Range[0, m - 1]! (* Jean-François Alcover, Sep 12 2022 *)

Prog

(PARI) a(n) = sum(k=0, n\2, (2*k)!*(k+1)^(k-1)*abs(stirling(n, 2*k, 1))/k!);
(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!)))
(PARI) my(N=30, x='x+O('x^N)); Vec(serlaplace(exp(-lambertw(-log(1-x)^2))))
(PARI) my(N=30, x='x+O('x^N)); Vec(serlaplace(-lambertw(-log(1-x)^2)/log(1-x)^2))

Crossrefs

Cf. A052813, A357027.

Sequence in context: A209738 A209459 A209521 * A366366 A357092 A132525

Adjacent sequences: A357023 A357024 A357025 * A357027 A357028 A357029

Keyword

nonn,changed

Author

Seiichi Manyama, Sep 09 2022

Status

approved