A340973 Generating function Sum_{n >= 0} a(n)*x^n = 1/sqrt((1-x)*(1-13*x)).

1, 7, 67, 721, 8179, 95557, 1137709, 13725439, 167204947, 2052215893, 25338173497, 314356676179, 3915672171229, 48938691421627, 613404577267843, 7707619156442401, 97058716523798227, 1224551690144551237

Offset

0,2

Links

Seiichi Manyama, Table of n, a(n) for n = 0..899

Formula

a(n) = Sum_{k=0..n} 3^k*binomial(n,k)*binomial(2*k,k).

a(n) = [x^n] (1+7*x+9*x^2)^n.

n * a(n) = 7 * (2*n-1) * a(n-1) - 13 * (n-1) * a(n-2) for n > 1.

E.g.f.: exp(7*x) * BesselI(0,6*x). - Ilya Gutkovskiy, Feb 01 2021

a(n) ~ 13^(n + 1/2) / (2 * sqrt(3*Pi*n)). - Vaclav Kotesovec, Nov 13 2021

Mathematica

a[n_] := Sum[3^k * Binomial[n, k] * Binomial[2*k, k], {k, 0, n}]; Array[a, 18, 0] (* Amiram Eldar, Feb 01 2021 *)
nxt[{n_, a_, b_}]:={n+1, b, (7*b(2n+1)-13*n*a)/(n+1)}; Join[{1}, NestList[nxt, {2, 7, 67}, 20] [[All, 2]]] (* Harvey P. Dale, Apr 27 2022 *)

Prog

(PARI) my(N=20, x='x+O('x^N)); Vec(1/sqrt((1-x)*(1-13*x)))
(PARI) {a(n) = sum(k=0, n, 3^k*binomial(n, k)*binomial(2*k, k))}
(PARI) {a(n) = polcoef((1+7*x+9*x^2)^n, n)}

Crossrefs

Column k=3 of A340970.

Cf. A322246, A337167.

Sequence in context: A073552 A036948 A020469 * A199756 A038386 A371398

Adjacent sequences: A340970 A340971 A340972 * A340974 A340975 A340976

Keyword

nonn

Author

Seiichi Manyama, Feb 01 2021

Status

approved