A337167 a(n) = 1 + 3 * Sum_{k=0..n-1} a(k) * a(n-k-1).

1, 4, 25, 199, 1795, 17422, 177463, 1870960, 20241403, 223438852, 2506596547, 28494103183, 327507800725, 3799735202218, 44440058006593, 523388751658831, 6201937444137619, 73888034816382820, 884517283667145259, 10634234680321209373, 128347834921058404249

Offset

0,2

Comments

Binomial transform of A005159.

Links

Seiichi Manyama, Table of n, a(n) for n = 0..901

N. J. A. Sloane, Transforms

Formula

G.f. A(x) satisfies: A(x) = 1 / (1 - x) + 3*x*A(x)^2.

G.f.: (1 - sqrt(1 - 12*x / (1 - x))) / (6*x).

E.g.f.: exp(7*x) * (BesselI(0,6*x) - BesselI(1,6*x)).

a(n) = Sum_{k=0..n} binomial(n,k) * 3^k * Catalan(k).

a(n) = 2F1([1/2, -n], [2], -12), where 2F1 is the hypergeometric function.

D-finite with recurrence (n+1) * a(n) = 2 * (7*n-3) * a(n-1) - 13 * (n-1) * a(n-2) for n > 1. - Seiichi Manyama, Jan 31 2021

a(n) ~ 13^(n + 3/2) / (8 * 3^(3/2) * sqrt(Pi) * n^(3/2)). - Vaclav Kotesovec, Feb 14 2021

Mathematica

a[n_] := a[n] = 1 + 3 Sum[a[k] a[n - k - 1], {k, 0, n - 1}]; Table[a[n], {n, 0, 20}]
Table[Sum[Binomial[n, k] 3^k CatalanNumber[k], {k, 0, n}], {n, 0, 20}]
Table[Hypergeometric2F1[1/2, -n, 2, -12], {n, 0, 20}]

Prog

(PARI) {a(n) = sum(k=0, n, 3^k*binomial(n, k)*(2*k)!/(k!*(k+1)!))} \\ Seiichi Manyama, Jan 31 2021
(PARI) my(N=20, x='x+O('x^N)); Vec(2/(1-x+sqrt((1-x)*(1-13*x)))) \\ Seiichi Manyama, Feb 01 2021

Crossrefs

Column k=3 of A340968.

Cf. A000108, A005159, A007317, A162326, A337169, A338979.

Sequence in context: A212694 A182953 A327074 * A140094 A284859 A144647

Adjacent sequences: A337164 A337165 A337166 * A337168 A337169 A337170

Keyword

nonn

Author

Ilya Gutkovskiy, Jan 28 2021

Status

approved