A349721 E.g.f. satisfies: A(x) = exp( x * (1 + 1/A(x)^3)/2 ).

1, 1, -2, 19, -260, 4966, -121328, 3613996, -127035920, 5147600680, -236245559984, 12112405259560, -686148484748480, 42560312499982720, -2868921992458611200, 208828244778853125376, -16324500711130356582656, 1363986660232205656646272

Offset: 0

Author

Seiichi Manyama, Nov 27 2021

Keywords

• sign

Links

• Seiichi Manyama, Table of n, a(n) for n = 0..352
• Eric Weisstein's World of Mathematics, Lambert W-Function.

Crossrefs

Cf. A007889, A202617, A349714, A349715, A349716, A349719, A349720.

Programs

Mathematica

a[n_] := (1/2^n) * Sum[(-3*k + 1)^(n - 1) * Binomial[n, k], {k, 0, n}]; Array[a, 18, 0] (* Amiram Eldar, Nov 27 2021 *)

PARI

a(n) = sum(k=0, n, (-3*k+1)^(n-1)*binomial(n, k))/2^n;

PARI

my(N=20, x='x+O('x^N)); Vec(serlaplace(((3*x/2)/lambertw(3*x/2*exp(-3*x/2)))^(1/3)))

PARI

my(N=20, x='x+O('x^N)); Vec(2*sum(k=0, N, (-3*k+1)^(k-1)*x^k/(2-(-3*k+1)*x)^(k+1)))

Formula

a(n) = (1/2^n) * Sum_{k=0..n} (-3*k+1)^(n-1) * binomial(n,k).

E.g.f.: ( (3*x/2)/LambertW( 3*x/2 * exp(-3*x/2) ) )^(1/3).

G.f.: 2 * Sum_{k>=0} (-3*k+1)^(k-1) * x^k/(2 - (-3*k+1)*x)^(k+1).

a(n) ~ -(-1)^n * sqrt(1 + LambertW(exp(-1))) * 3^(n-1) * n^(n-1) / (2^n * exp(n) * LambertW(exp(-1))^(n - 1/3)). - Vaclav Kotesovec, Dec 05 2021