A356906 E.g.f. satisfies A(x)^(A(x)^2) = 1/(1 - x)^x.

1, 0, 2, 3, -28, -150, 2154, 26040, -322512, -7872984, 77570280, 3752301960, -22068935736, -2542757920560, 1422846762960, 2302464947491800, 14860063644794880, -2653728770258072640, -41790782141846648640, 3739260018343338345600

Offset

0,3

Links

Table of n, a(n) for n=0..19.

Formula

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

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

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

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

Prog

(PARI) a(n) = n!*sum(k=0, n\2, (-2*k+1)^(k-1)*abs(stirling(n-k, k, 1))/(n-k)!);
(PARI) my(N=20, x='x+O('x^N)); Vec(serlaplace(sum(k=0, N, (-2*k+1)^(k-1)*(-x*log(1-x))^k/k!)))
(PARI) my(N=20, x='x+O('x^N)); Vec(serlaplace(exp(lambertw(-2*x*log(1-x))/2)))
(PARI) my(N=20, x='x+O('x^N)); Vec(serlaplace((-2*x*log(1-x)/lambertw(-2*x*log(1-x)))^(1/2)))

Crossrefs

Cf. A355842, A356795, A356796, A356905.

Cf. A349652, A356885.

Sequence in context: A357267 A074233 A354611 * A371115 A206592 A126266

Adjacent sequences: A356903 A356904 A356905 * A356907 A356908 A356909

Keyword

sign

Author

Seiichi Manyama, Sep 03 2022

Status

approved