A386844 a(n) = Sum_{k=0..n} binomial(3*n+2,k) * binomial(3*n-k,n-k).
1, 8, 83, 942, 11177, 136164, 1688031, 21187546, 268409813, 3424751568, 43948343243, 566607282118, 7333422759873, 95225755205564, 1239995365588919, 16186010348814258, 211729232160358317, 2774813844884684712, 36425708310248816547, 478880147399497482142, 6304133921156502650777
Offset: 0
Keywords
Links
- Vincenzo Librandi, Table of n, a(n) for n = 0..350
Programs
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Magma
[&+[Binomial(3*n+2,k) * Binomial(3*n-k,n-k): k in [0..n]]: n in [0..25]]; // Vincenzo Librandi, Aug 06 2025
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Mathematica
Table[Sum[Binomial[3*n+2,k]*Binomial[3*n-k,n-k],{k,0,n}],{n,0,25}] (* Vincenzo Librandi, Aug 06 2025 *) -
PARI
a(n) = sum(k=0, n, binomial(3*n+2, k)*binomial(3*n-k, n-k));
Formula
a(n) = [x^n] (1+x)^(3*n+2)/(1-x)^(2*n+1).
a(n) = [x^n] 1/((1-x)^2 * (1-2*x)^(2*n+1)).
a(n) = Sum_{k=0..n} 2^k * (-1)^(n-k) * (n-k+1) * binomial(3*n+2,k).
a(n) = Sum_{k=0..n} 2^k * (n-k+1) * binomial(2*n+k,k).
a(n) ~ 3^(3*n + 5/2) / (25 * sqrt(Pi*n) * 2^(n-1)). - Vaclav Kotesovec, Aug 07 2025