A386920 a(n) = Sum_{k=0..n} binomial(4*n,k) * binomial(3*n-k,n-k).
1, 7, 83, 1102, 15395, 221402, 3244430, 48173244, 722264355, 10910288290, 165788618138, 2531447611524, 38807906496398, 596945491933252, 9208704207465020, 142410375212008952, 2207122379129757987, 34272045530904650610, 533075544700619580002, 8304126391210396590900
Offset: 0
Keywords
Links
- Vincenzo Librandi, Table of n, a(n) for n = 0..350
Programs
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Magma
[&+[Binomial(4*n,k) * Binomial(3*n-k,n-k): k in [0..n]]: n in [0..25]]; // Vincenzo Librandi, Aug 10 2025
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Mathematica
Table[Sum[Binomial[4*n,k]*Binomial[3*n-k,n-k],{k,0,n}],{n,0,25}] (* Vincenzo Librandi, Aug 10 2025 *) -
PARI
a(n) = sum(k=0, n, binomial(4*n, k)*binomial(3*n-k, n-k));
Formula
a(n) = [x^n] (1+x)^(4*n)/(1-x)^(2*n+1).
a(n) = [x^n] 1/((1-x)^n * (1-2*x)^(2*n+1)).
a(n) = Sum_{k=0..n} 2^k * (-1)^(n-k) * binomial(4*n,k) * binomial(2*n-k-1,n-k).
a(n) = Sum_{k=0..n} 2^k * binomial(2*n+k,k) * binomial(2*n-k-1,n-k).
a(n) ~ (2 + sqrt(2)) * 2^(4*n-2) / sqrt(Pi*n). - Vaclav Kotesovec, Aug 21 2025