A349714 E.g.f. satisfies: A(x) = exp( x * (1 + A(x)^3)/2 ).

1, 1, 4, 37, 532, 10426, 259300, 7823908, 277713904, 11339452792, 523621438336, 26982030104536, 1534947906550528, 95550736737542464, 6460746383585984512, 471533064029919744256, 36946948091091750496000, 3093472887944746070621056

Offset

0,3

Links

Seiichi Manyama, Table of n, a(n) for n = 0..352

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

Formula

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

E.g.f.: ( -LambertW( -3*x/2 * exp(3*x/2) )/(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) ~ sqrt(1 + LambertW(exp(-1))) * 3^(n-1) * n^(n-1) / (LambertW(exp(-1))^(n + 1/3) * 2^n * exp(n)). - Vaclav Kotesovec, Nov 26 2021

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 *)

Prog

(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((-lambertw(-3*x/2*exp(3*x/2))/(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)))

Crossrefs

Cf. A007889, A202617, A349715, A349716, A349719, A349720, A349721.

Sequence in context: A365778 A277638 A352237 * A121080 A001518 A185082

Adjacent sequences: A349711 A349712 A349713 * A349715 A349716 A349717

Keyword

nonn

Author

Seiichi Manyama, Nov 26 2021

Status

approved