A387276 - OEIS

%I #17 Aug 31 2025 10:46:44

%S 1,20,255,2650,24521,210840,1725234,13631700,104993955,793367300,

%T 5907885412,43495473840,317355930255,2298888740400,16555878011448,

%U 118661449810320,847132614218907,6027874235210700,42773816956415055,302816249208061050,2139537520524710691

%N a(n) = Sum_{k=0..n} 3^(n-k) * binomial(n+3,k+3) * binomial(2*k+6,k+6).

%H Vincenzo Librandi, <a href="/A387276/b387276.txt">Table of n, a(n) for n = 0..800</a>

%F n*(n+6)*a(n) = (n+3) * (5*(2*n+5)*a(n-1) - 21*(n+2)*a(n-2)) for n > 1.

%F a(n) = Sum_{k=0..floor(n/2)} 5^(n-2*k) * binomial(n+3,n-2*k) * binomial(2*k+3,k).

%F a(n) = [x^n] (1+5*x+x^2)^(n+3).

%F E.g.f.: exp(5*x) * BesselI(3, 2*x), with offset 3.

%t Table[Sum[3^(n-k)*Binomial[n+3,k+3]*Binomial[2*k+6,k+6],{k,0,n}],{n,0,25}] (* _Vincenzo Librandi_, Aug 31 2025 *)

%o (PARI) a(n) = sum(k=0, n, 3^(n-k)*binomial(n+3, k+3)*binomial(2*k+6, k+6));

%o (Magma) [&+[3^(n-k) * Binomial(n+3,k+3) * Binomial(2*k+6,k+6): k in [0..n]]: n in [0..25]]; // _Vincenzo Librandi_, Aug 31 2025

%Y Cf. A387275, A387277, A387278.

%Y Cf. A387239, A387273.

%K nonn

%O 0,2

%A _Seiichi Manyama_, Aug 24 2025