A345103 a(n) = 1 + 4 * Sum_{k=0..n-1} binomial(n,k) * a(k) * a(n-k-1).

1, 5, 61, 1277, 37741, 1437725, 67013101, 3693540317, 234974905261, 16945434018845, 1366008048556141, 121721015465713757, 11880107754103150381, 1260413749895624939165, 144427420001275864755181, 17776090894283922227621597, 2338833689096321086977341101, 327585830473259220341296486685

Offset: 0

Author

Ilya Gutkovskiy, Jun 08 2021

Keywords

• nonn

Links

• Seiichi Manyama, Table of n, a(n) for n = 0..331

Crossrefs

Cf. A006677, A052886, A144828, A201365, A345102.

Programs

Mathematica

a[n_] := a[n] = 1 + 4 Sum[Binomial[n, k] a[k] a[n - k - 1], {k, 0, n - 1}]; Table[a[n], {n, 0, 17}]
nmax = 17; CoefficientList[Series[Exp[x]/Sqrt[9 - 8 Exp[x]], {x, 0, nmax}], x] Range[0, nmax]!
Table[Sum[Sum[Binomial[n, k] StirlingS2[k, j] 4^j (2 j - 1)!!, {j, 0, k}], {k, 0, n}], {n, 0, 17}]

PARI

N=20; x='x+O('x^N); Vec(serlaplace(exp(x)/sqrt(9-8*exp(x)))) \\ Seiichi Manyama, Oct 20 2021

Formula

E.g.f.: exp(x) / sqrt(9 - 8 * exp(x)).