A385514 a(n) = Sum_{k=0..n} 2^(n-k) * binomial(2*n+1,k) * binomial(2*n-k,n-k).
1, 7, 64, 643, 6766, 73162, 805414, 8979523, 101060326, 1145704162, 13064219224, 149674343518, 1721537039236, 19866626222632, 229912254620434, 2667252458378083, 31009548579437446, 361198085246048602, 4214267651960927824, 49243413868632029338, 576179701092650156356
Offset: 0
Keywords
Links
- Vincenzo Librandi, Table of n, a(n) for n = 0..350
Programs
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Magma
[&+[2^(n-k) * Binomial(2*n+1,k) * Binomial(2*n-k,n-k): k in [0..n]]: n in [0..25]]; // Vincenzo Librandi, Aug 05 2025
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Mathematica
Table[Sum[2^(n-k)*Binomial[2*n+1,k]*Binomial[2*n-k,n-k],{k,0,n}],{n,0,25}] (* Vincenzo Librandi, Aug 05 2025 *) -
PARI
a(n) = sum(k=0, n, 2^(n-k)*binomial(2*n+1, k)*binomial(2*n-k, n-k));
Formula
a(n) = [x^n] (1+x)^(2*n+1)/(1-2*x)^(n+1).
a(n) = [x^n] 1/((1-x) * (1-3*x)^(n+1)).
a(n) = Sum_{k=0..n} 3^k * (-2)^(n-k) * binomial(2*n+1,k).
a(n) = Sum_{k=0..n} 3^k * binomial(n+k,k).
a(n) ~ 2^(2*n+1) * 3^(n+1) / (5*sqrt(Pi*n)). - Vaclav Kotesovec, Aug 06 2025