A386958 a(n) = Sum_{k=0..n} 8^k * binomial(k-2/3,k) * binomial(2*n+1/3,n-k).

1, 5, 33, 248, 2020, 17325, 153699, 1395084, 12868839, 120127865, 1131633217, 10737438816, 102480890512, 982880111192, 9465545374920, 91479218990688, 886803360846876, 8619761335490460, 83982810424366860, 819973263265010400, 8020986875021209320

Offset: 0

Author

Seiichi Manyama, Aug 11 2025

Keywords

• nonn
• changed

Links

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

Crossrefs

Cf. A386957.

Programs

Mathematica

Table[Sum[8^k*Binomial[k-2/3,k]*Binomial[2*n+1/3, n-k],{k,0,n}],{n,0,25}] (* Vincenzo Librandi, Sep 03 2025 *)

PARI

a(n) = sum(k=0, n, 8^k*binomial(k-2/3, k)*binomial(2*n+1/3, n-k));

Formula

a(n) = [x^n] 1/((1-9*x)^(1/3) * (1-x)^(n+1)).

a(n) = Sum_{k=0..n} 9^k * (-8)^(n-k) * binomial(2*n+1/3,k) * binomial(2*n-k,n-k).

a(n) = Sum_{k=0..n} 9^k * binomial(k-2/3,k) * binomial(2*n-k,n-k).

G.f.: 1/( sqrt(1-4*x) * ((9*sqrt(1-4*x)-7)/2)^(1/3) ).