A349719 E.g.f. satisfies: A(x) = exp( x * (1 + 1/A(x))/2 ).

1, 1, 0, 1, -4, 26, -212, 2108, -24720, 334072, -5112544, 87396728, -1650607040, 34132685120, -767025716736, 18612106195456, -485013257865472, 13509071081429888, -400505695457942528, 12592502771190979712, -418524228123134068224

Offset

0,5

Links

Seiichi Manyama, Table of n, a(n) for n = 0..411

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

Formula

a(n) = (1/2^n) * Sum_{k=0..n} (-k+1)^(n-1) * binomial(n,k).

E.g.f.: (x/2)/LambertW( x/2 * exp(-x/2) ).

G.f.: 2 * Sum_{k>=0} (-k+1)^(k-1) * x^k/(2 - (-k+1)*x)^(k+1).

a(n) ~ -(-1)^n * sqrt(1 + LambertW(exp(-1))) * n^(n-1) / (2^n * exp(n) * LambertW(exp(-1))^(n-1)). - Vaclav Kotesovec, Dec 05 2021

Mathematica

a[n_] := (1/2^n) * Sum[If[k == n == 1, 1, (-k + 1)^(n - 1)] * Binomial[n, k], {k, 0, n}]; Array[a, 21, 0] (* Amiram Eldar, Nov 27 2021 *)

Prog

(PARI) a(n) = sum(k=0, n, (-k+1)^(n-1)*binomial(n, k))/2^n;
(PARI) my(N=40, x='x+O('x^N)); Vec(serlaplace((x/2)/lambertw(x/2*exp(-x/2))))
(PARI) my(N=40, x='x+O('x^N)); Vec(2*sum(k=0, N, (-k+1)^(k-1)*x^k/(2-(-k+1)*x)^(k+1)))

Crossrefs

Cf. A007889, A202617, A283828, A349714, A349715, A349716, A349720, A349721.

Sequence in context: A239295 A274735 A355379 * A283828 A306028 A164101

Adjacent sequences: A349716 A349717 A349718 * A349720 A349721 A349722

Keyword

sign

Author

Seiichi Manyama, Nov 27 2021

Status

approved