A386362 Expansion of (1/x) * Series_Reversion( x/(1+7*x+9*x^2) ).

1, 7, 58, 532, 5209, 53347, 564499, 6123481, 67732483, 761052565, 8662502212, 99671232514, 1157409133831, 13546774268125, 159649564550746, 1892849564159596, 22562032457415067, 270209749616920813, 3249905798884688038, 39237866746912398292, 475388228365424562019

Offset: 0

Author

Seiichi Manyama, Aug 20 2025

Keywords

• nonn
• new

Crossrefs

Column k=3 of A386408.

Cf. A002212, A127846, A386389.

Cf. A000108, A337167, A385716.

Programs

PARI

my(N=30, x='x+O('x^N)); Vec(serreverse(x/(1+7*x+9*x^2))/x)

Formula

G.f.: 2/(1 - 7*x + sqrt((1-x) * (1-13*x))).

a(n) = (A337167(n+1) - A337167(n))/3.

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

a(n) = Sum_{k=0..floor(n/2)} 9^k * 7^(n-2*k) * binomial(n,2*k) * Catalan(k).

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