A385639 a(n) = Sum_{k=0..n} binomial(4*n+1,k) * binomial(2*n-k,n-k).
1, 7, 69, 748, 8485, 98847, 1171884, 14066808, 170421669, 2079531685, 25520363869, 314653207128, 3894577133356, 48362609654548, 602248101550920, 7517853111444528, 94044248726758821, 1178641094940246897, 14796230460187072719, 186022053254555479500, 2341837809478393341885
Offset: 0
Keywords
Programs
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Mathematica
Table[Sum[Binomial[4*n+1, k]*Binomial[2*n-k, n-k], {k, 0, n}], {n, 0, 20}] (* Vaclav Kotesovec, Aug 07 2025 *) -
PARI
a(n) = sum(k=0, n, binomial(4*n+1, k)*binomial(2*n-k, n-k));
Formula
a(n) = [x^n] (1+x)^(4*n+1)/(1-x)^(n+1).
a(n) = [x^n] 1/((1-x)^(2*n+1) * (1-2*x)^(n+1)).
a(n) = Sum_{k=0..n} 2^k * (-1)^(n-k) * binomial(4*n+1,k) * binomial(3*n-k,n-k).
a(n) = Sum_{k=0..n} 2^k * binomial(n+k,k) * binomial(3*n-k,n-k).
a(n) = binomial(2*n, n)*hypergeom([-1-4*n, -n], [-2*n], -1). - Stefano Spezia, Aug 07 2025
a(n) ~ sqrt((187 - 3*sqrt(17)) / (17*Pi*n)) * (51*sqrt(17) - 107)^n / 2^(3*n + 3/2). - Vaclav Kotesovec, Aug 07 2025