A387272 a(n) = Sum_{k=0..n} 2^(n-k) * binomial(n+2,k+2) * binomial(2*k+4,k+4).
1, 12, 100, 720, 4815, 30884, 193144, 1188576, 7236690, 43741720, 263056728, 1576298464, 9421080123, 56200937940, 334801389360, 1992471776448, 11848869296622, 70425535830696, 418426332826200, 2485390365370080, 14760336569524854, 87650482093915752
Offset: 0
Links
- Vincenzo Librandi, Table of n, a(n) for n = 0..800
Programs
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Magma
[&+[2^(n-k) * Binomial(n+2,k+2) * Binomial(2*k+4,k+4): k in [0..n]]: n in [0..25]]; // Vincenzo Librandi, Aug 31 2025
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Mathematica
Table[Sum[2^(n-k)*Binomial[n+2,k+2]*Binomial[2*k+4,k+4],{k,0,n}],{n,0,25}] (* Vincenzo Librandi, Aug 31 2025 *) -
PARI
a(n) = sum(k=0, n, 2^(n-k)*binomial(n+2, k+2)*binomial(2*k+4, k+4));
Formula
n*(n+4)*a(n) = (n+2) * (4*(2*n+3)*a(n-1) - 12*(n+1)*a(n-2)) for n > 1.
a(n) = Sum_{k=0..floor(n/2)} 4^(n-2*k) * binomial(n+2,n-2*k) * binomial(2*k+2,k).
a(n) = [x^n] (1+4*x+x^2)^(n+2).
E.g.f.: exp(4*x) * BesselI(2, 2*x), with offset 2.