A387239 a(n) = Sum_{k=0..n} binomial(n+3,k+3) * binomial(2*k+6,k+6).
1, 12, 95, 630, 3801, 21672, 119154, 639180, 3369795, 17543196, 90476100, 463291920, 2359240975, 11961944400, 60440659640, 304543085040, 1531044995355, 7682898791700, 38494752520175, 192632866196694, 962948703201331, 4809438625979592, 24002988378037350, 119719958370912900
Offset: 0
Links
- Vincenzo Librandi, Table of n, a(n) for n = 0..500
Programs
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Magma
[&+[Binomial(n+3,k+3) * Binomial(2*k+6,k+6): k in [0..n]]: n in [0..25]]; // Vincenzo Librandi, Aug 24 2025
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Mathematica
Table[Sum[Binomial[n+3,k+3]* Binomial[2*k+6, k+6],{k,0,n}],{n,0,25}] (* Vincenzo Librandi, Aug 24 2025 *) -
PARI
a(n) = sum(k=0, n, binomial(n+3, k+3)*binomial(2*k+6, k+6));
Formula
n*(n+6)*a(n) = (n+3) * (3*(2*n+5)*a(n-1) - 5*(n+2)*a(n-2)) for n > 1.
a(n) = Sum_{k=0..floor(n/2)} 3^(n-2*k) * binomial(n+3,n-2*k) * binomial(2*k+3,k).
a(n) = [x^n] (1+3*x+x^2)^(n+3).
E.g.f.: exp(3*x) * BesselI(3, 2*x), with offset 3.