A386719 a(n) = Sum_{k=0..n} 2^k * binomial(4*n,k) * binomial(4*n-k-1,n-k).
1, 11, 229, 5381, 133333, 3404156, 88600483, 2337160718, 62263902037, 1671407550260, 45137852641204, 1224954657942125, 33377579214681619, 912572183952374996, 25023054179816358034, 687862647149533181036, 18950129471489195622229, 523067259899842250453060
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Keywords
Programs
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Mathematica
Table[Sum[3^k*(-1)^(n-k)*Binomial[4*n, k], {k, 0, n}], {n, 0, 25}] (* Vaclav Kotesovec, Jul 31 2025 *) -
PARI
a(n) = sum(k=0, n, 2^k*binomial(4*n, k)*binomial(4*n-k-1, n-k));
Formula
a(n) = [x^n] ( (1+2*x)^4/(1-x)^3 )^n.
The g.f. exp( Sum_{k>=1} a(k) * x^k/k ) has integer coefficients and equals (1/x) * Series_Reversion( x * (1-x)^3 / (1+2*x)^4 ). See A386723.
a(n) = Sum_{k=0..n} 3^k * (-1)^(n-k) * binomial(4*n,k).
a(n) ~ 2^(8*n - 1/2) / (5 * sqrt(Pi*n) * 3^(2*n - 3/2)). - Vaclav Kotesovec, Jul 31 2025
a(n) = Sum_{k=0..n} 3^k * 2^(n-k) * binomial(3*n+k-1,k). - Seiichi Manyama, Aug 01 2025