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

Author

Seiichi Manyama, Feb 01 2021

Keywords

• nonn

Links

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

Crossrefs

Column k=3 of A340970.

Cf. A322246, A337167.

Programs

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 *)

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);

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

From Seiichi Manyama, Aug 19 2025: (Start)

a(n) = (1/4)^n * Sum_{k=0..n} 13^k * binomial(2*k,k) * binomial(2*(n-k),n-k).

a(n) = Sum_{k=0..n} (-3)^k * 13^(n-k) * binomial(n,k) * binomial(2*k,k). (End)