A386865 a(n) = Sum_{k=0..n} 2^(n-k) * binomial(2*n+2,k) * binomial(2*n-k-1,n-k).
1, 6, 51, 496, 5130, 54894, 600103, 6657312, 74646702, 843819580, 9599776494, 109776491664, 1260666279964, 14528980409454, 167951183468655, 1946529575164864, 22611104963042646, 263175370423429428, 3068541416792813338, 35834296592951011680, 419059482092284948908
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Keywords
Programs
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Mathematica
Table[Sum[(-1)^k*(k+1)*(k+2)*2^(k-1)*3^(n-k)* Binomial[2*n+2, n+k+2], {k, 0, n}], {n, 0, 20}] (* Vaclav Kotesovec, Aug 06 2025 *) -
PARI
a(n) = sum(k=0, n, 2^(n-k)*binomial(2*n+2, k)*binomial(2*n-k-1, n-k));
Formula
a(n) = [x^n] (1+x)^(2*n+2)/(1-2*x)^n.
a(n) = [x^n] 1/((1-x)^3 * (1-3*x)^n).
a(n) = Sum_{k=0..n} 3^k * (-2)^(n-k) * binomial(2*n+2,k) * binomial(n-k+2,n-k).
a(n) = Sum_{k=0..n} 3^k * binomial(n+k-1,k) * binomial(n-k+2,n-k).
a(n) ~ 2^(2*n+2) * 3^(n+3) / (125*sqrt(Pi*n)). - Vaclav Kotesovec, Aug 06 2025