A345102 -id:A345102 - cp's OEIS frontend

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

Ilya Gutkovskiy, Jun 08 2021

nonn

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

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

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}]

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

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

A345105 a(n) = 1 + 3 * Sum_{k=0..n-1} binomial(n-1,k) * a(k) * a(n-k-1).

Original entry on oeis.org

1, 4, 25, 247, 3283, 54661, 1092427, 25473037, 678837319, 20351864821, 677954261635, 24842157250117, 993040102321927, 43003754679356941, 2005536858420616963, 100211634039201328381, 5341144936822423446247, 302468060262966258380773, 18136282125753572653056355

Offset: 0

Ilya Gutkovskiy, Jun 08 2021

nonn

Cf. A032031, A054687, A345102, A345104.

a[n_] := a[n] = 1 + 3 Sum[Binomial[n - 1, k] a[k] a[n - k - 1], {k, 0, n - 1}]; Table[a[n], {n, 0, 18}]
nmax = 18; A[] = 1; Do[A[x] = Normal[Integrate[3 A[x]^2 + Exp[x], x] + O[x]^(nmax + 1)], nmax]; CoefficientList[A[x], x] Range[0, nmax]!

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

cp's OEIS Frontend

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

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