A361067 E.g.f. satisfies A(x) = exp( x/((1-x) * A(x)) ).

1, 1, 1, 4, 9, 76, 175, 3606, 833, 354376, -1605249, 65111410, -718371071, 20105327100, -351241054177, 9362931464446, -214514949732735, 6039303900168976, -165679758877120001, 5093296357218337386, -159900268661169533119, 5405435526807425433220

Offset

0,4

Links

Winston de Greef, Table of n, a(n) for n = 0..413

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

Formula

a(n) = n! * Sum_{k=0..n} (-k+1)^(k-1) * binomial(n-1,n-k)/k!.

E.g.f.: exp( LambertW(x/(1-x)) ).

E.g.f.: 1 / ( (1-x)/x * LambertW(x/(1-x)) ).

a(n) ~ -(-1)^n * exp(-1) * (1 - exp(-1))^(n + 1/2) * n^(n-1). - Vaclav Kotesovec, Mar 02 2023

Mathematica

nmax = 21; A[_] = 1;
Do[A[x_] = Exp[x/((1 - x)*A[x])] + O[x]^(nmax+1) // Normal, {nmax}];
CoefficientList[A[x], x]*Range[0, nmax]! (* Jean-François Alcover, Mar 04 2024 *)

Prog

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

Crossrefs

Cf. A052868, A361065, A361066, A361068, A361069.

Sequence in context: A327579 A041597 A041030 * A359654 A061104 A082381

Adjacent sequences: A361064 A361065 A361066 * A361068 A361069 A361070

Keyword

sign

Author

Seiichi Manyama, Mar 01 2023

Status

approved