A349720 E.g.f. satisfies: A(x) = exp( x * (1 + 1/A(x)^2)/2 ).

1, 1, -1, 7, -63, 801, -13025, 258343, -6048511, 163276417, -4992740289, 170571634311, -6439161507647, 266180947507489, -11958385377911713, 580151397382158631, -30227616424300542975, 1683438461080186841601, -99796591057813372007297

Offset

0,4

Links

Seiichi Manyama, Table of n, a(n) for n = 0..372

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

Formula

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

E.g.f.: ( x/LambertW( x * exp(-x) ) )^(1/2).

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

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

Mathematica

a[n_] := (1/2^n) * Sum[(-2*k + 1)^(n - 1) * Binomial[n, k], {k, 0, n}]; Array[a, 19, 0] (* Amiram Eldar, Nov 27 2021 *)

Prog

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

Crossrefs

Cf. A007889, A202617, A349714, A349715, A349716, A349719, A349721.

Sequence in context: A346683 A084063 A184141 * A152797 A345383 A126883

Adjacent sequences: A349717 A349718 A349719 * A349721 A349722 A349723

Keyword

sign

Author

Seiichi Manyama, Nov 27 2021

Status

approved