A386389 Expansion of (1/x) * Series_Reversion( x/(1+9*x+16*x^2) ).

1, 9, 97, 1161, 14849, 198729, 2748641, 38977353, 563644673, 8280210825, 123226850913, 1853870946057, 28148395838721, 430791367720905, 6638484468424929, 102918165951351753, 1604104541561284097, 25121009971212463881, 395085505395126968417, 6237523016309454855561

Offset

0,2

Links

Vincenzo Librandi, Table of n, a(n) for n = 0..800

Formula

G.f.: 2/(1 - 9*x + sqrt((1-x) * (1-17*x))).

a(n) = (A386387(n+1) - A386387(n))/4.

(n+2)*a(n) = 9*(2*n+1)*a(n-1) - 17*(n-1)*a(n-2) for n > 1.

a(n) = Sum_{k=0..floor(n/2)} 16^k * 9^(n-2*k) * binomial(n,2*k) * Catalan(k).

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

E.g.f.: exp(9*x)*BesselI(1, 8*x)/(4*x). - Stefano Spezia, Oct 30 2025

Mathematica

Table[Sum[ 16^k*9^(n-2*k)*Binomial[n, 2*k]*CatalanNumber[k], {k, 0, Floor[n/2]}], {n, 0, 30}] (* Vincenzo Librandi, Oct 30 2025 *)

Prog

(PARI) my(N=30, x='x+O('x^N)); Vec(serreverse(x/(1+9*x+16*x^2))/x)
(Magma) [&+[Catalan(k)*16^k * 9^(n-2*k)* Binomial(n, 2*k): k in [0..Floor(n/2)]] : n in [0..30] ]; // Vincenzo Librandi, Oct 30 2025

Crossrefs

Column k=4 of A386408.

Cf. A002212, A127846, A386362.

Cf. A000108, A269796, A386387.

Sequence in context: A180675 A083077 A194725 * A218500 A293986 A380639

Adjacent sequences: A386386 A386387 A386388 * A386390 A386391 A386392

Keyword

nonn

Author

Seiichi Manyama, Aug 20 2025

Status

approved