A260902 a(n) = (-5*(-1)^n + Sum_{k>=0} 2*A000108(k)*k^n/5^k)/sqrt(5), where A000108 are Catalan numbers.

-1, 3, 1, 27, 289, 4683, 95761, 2382747, 69870529, 2359997163, 90239163121, 3853391348667, 181765659243169, 9386568200722443, 526713953100688081, 31912283163641549787, 2076293242327577102209, 144382074825232693748523, 10686433228580787658046641

Offset: 0

Author

Vaclav Kotesovec, Nov 17 2015

Keywords

• sign

Links

• Vaclav Kotesovec, Table of n, a(n) for n = 0..100

Crossrefs

Cf. A000108, A260701, A348468.

Programs

Mathematica

Table[(-5*(-1)^n + Sum[2*CatalanNumber[k] k^n/5^k, {k, 0, Infinity}]) / Sqrt[5], {n, 0, 20}]

PARI

N=20; x='x+O('x^N); Vec(serlaplace(-sqrt(exp(-x)*(-4+5*exp(-x))))) \\ Seiichi Manyama, Oct 21 2021

Formula

a(n) ~ 2^(3/2) * n^(n-1) / (sqrt(5) * exp(n) * log(5/4)^(n-1/2)).

E.g.f.: -sqrt( exp(-x) * (-4+5*exp(-x)) ). - Seiichi Manyama, Oct 21 2021