A386941 a(n) = Sum_{k=0..n} (2*k+1) * binomial(2*k,k) * binomial(2*n-k-1,n-k).
1, 7, 45, 276, 1645, 9618, 55468, 316620, 1792989, 10089420, 56482998, 314859636, 1748876220, 9684449908, 53487036420, 294732771280, 1620825793053, 8897604701130, 48766676365204, 266905699036900, 1458941915879910, 7965552023094600, 43444688665988700
Offset: 0
Programs
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PARI
a(n) = sum(k=0, n, (2*k+1)*binomial(2*k, k)*binomial(2*n-k-1, n-k));
Formula
a(n) = [x^n] 1/((1-4*x)^(3/2) * (1-x)^n).
G.f.: (1+sqrt(1-4*x))/sqrt( 4 * (1-4*x) * (2*sqrt(1-4*x)-1)^3 ).
a(n) = Sum_{k=0..n} 3^(n-k) * binomial(2*n+1/2,k) * binomial(n-k+1/2,n-k) = Sum_{k=0..n} (2*k+1) * (3/4)^k * binomial(2*k,k) * binomial(2*n+1/2,n-k).
a(n) = Sum_{k=0..n} 4^k * (-3)^(n-k) * binomial(2*n+1/2,k) * binomial(2*n-k-1,n-k).