A385319 a(n) = Sum_{k=0..n} 2^k * binomial(2*n,k) * binomial(2*n-k-1,n-k).
1, 5, 43, 422, 4387, 47090, 515854, 5731052, 64330531, 727812026, 8285505178, 94798502804, 1089146648206, 12556967516852, 145201851788092, 1683334752235352, 19558532125813027, 227694254392461962, 2655343386035416162, 31014205667706302852, 362746369474101224602
Offset: 0
Keywords
Programs
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Mathematica
Table[Sum[3^k*(-1)^(n-k)*Binomial[2*n, k], {k, 0, n}], {n, 0, 20}] (* Vaclav Kotesovec, Jul 31 2025 *) -
PARI
a(n) = sum(k=0, n, 2^k*binomial(2*n,k)*binomial(2*n-k-1, n-k));
Formula
a(n) = [x^n] ( (1+2*x)^2/(1-x) )^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) / (1+2*x)^2 ). See A371391.
a(n) = Sum_{k=0..n} 3^k * (-1)^(n-k) * binomial(2*n,k).
a(n) ~ 2^(2*n-2) * 3^(n+1) / sqrt(Pi*n). - Vaclav Kotesovec, Jul 31 2025
a(n) = Sum_{k=0..n} 3^k * 2^(n-k) * binomial(n+k-1,k). - Seiichi Manyama, Aug 01 2025