A345103 - OEIS

%I #9 Oct 20 2021 07:49:35

%S 1,5,61,1277,37741,1437725,67013101,3693540317,234974905261,

%T 16945434018845,1366008048556141,121721015465713757,

%U 11880107754103150381,1260413749895624939165,144427420001275864755181,17776090894283922227621597,2338833689096321086977341101,327585830473259220341296486685

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

%H Seiichi Manyama, <a href="/A345103/b345103.txt">Table of n, a(n) for n = 0..331</a>

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

%t 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}]

%t nmax = 17; CoefficientList[Series[Exp[x]/Sqrt[9 - 8 Exp[x]], {x, 0, nmax}], x] Range[0, nmax]!

%t Table[Sum[Sum[Binomial[n, k] StirlingS2[k, j] 4^j (2 j - 1)!!, {j, 0, k}], {k, 0, n}], {n, 0, 17}]

%o (PARI) N=20; x='x+O('x^N); Vec(serlaplace(exp(x)/sqrt(9-8*exp(x)))) \\ _Seiichi Manyama_, Oct 20 2021

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

%K nonn

%O 0,2

%A _Ilya Gutkovskiy_, Jun 08 2021