A356795 E.g.f. satisfies A(x) = 1/(1 - x)^(x * A(x)^2).

1, 0, 2, 3, 68, 330, 7674, 73080, 1883440, 28281960, 818625960, 17120406600, 557507325000, 15014517495120, 548643259812816, 18056683281775320, 736892260092195840, 28579282973977498560, 1295028345251832359616, 57666859088090317591680

Offset: 0

Author

Seiichi Manyama, Aug 28 2022

Keywords

• nonn

Links

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

Crossrefs

Cf. A066166, A355842, A356796.

Cf. A356786.

Programs

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((lambertw(2*x*log(1-x))/(2*x*log(1-x)))^(1/2)))

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) = ( LambertW(2 * x * log(1-x))/(2 * x * log(1-x)) )^(1/2).