A386960 a(n) = Sum_{k=0..n} 8^k * binomial(2*n,n-k).

1, 10, 102, 1036, 10502, 106380, 1077276, 10908096, 110447046, 1118286172, 11322685172, 114642332232, 1160754172316, 11752638152824, 118995469654968, 1204829162684136, 12198895398209862, 123513816397462524, 1250577392936568708, 12662096110945862856, 128203723152486704052

Offset: 0

Author

Seiichi Manyama, Aug 11 2025

Keywords

• nonn

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..400

Crossrefs

Cf. A032443, A072547, A088218, A100192, A100193.

Programs

Magma

[&+[8^k * Binomial(2*n, n-k): k in [0..n]]: n in [0..25]]; // Vincenzo Librandi, Aug 13 2025

Mathematica

Table[Sum[8^k*Binomial[2*n,n-k],{k,0,n}],{n,0,25}] (* Vincenzo Librandi, Aug 13 2025 *)

PARI

a(n) = sum(k=0, n, 8^k*binomial(2*n, n-k));

Formula

a(n) = [x^n] 1/((1-9*x) * (1-x)^n).

a(n) = Sum_{k=0..n} 9^k * (-8)^(n-k) * binomial(2*n,k) * binomial(2*n-k-1,n-k).

a(n) = Sum_{k=0..n} 9^k * binomial(2*n-k-1,n-k).

G.f.: (1+sqrt(1-4*x))/( sqrt(1-4*x) * (9*sqrt(1-4*x)-7) ).