cp's OEIS Frontend

A387308 a(n) = Sum_{k=0..n} 2^k * binomial(n+3,k+3) * binomial(2*k+6,k+6).

%I A387308 #15 Aug 30 2025 10:26:21
%S A387308 1,20,270,3100,32711,328440,3195276,30433800,285604605,2651696300,
%T A387308 24425110138,223636254660,2038173128355,18508562948400,
%U A387308 167596683043032,1514149108652880,13654464563944377,122951462526317700,1105768912324277670,9934852156019798700,89186900539764803391
%N A387308 a(n) = Sum_{k=0..n} 2^k * binomial(n+3,k+3) * binomial(2*k+6,k+6).
%H A387308 Vincenzo Librandi, <a href="/A387308/b387308.txt">Table of n, a(n) for n = 0..800</a>
%F A387308 n*(n+6)*a(n) = (n+3) * (5*(2*n+5)*a(n-1) - 9*(n+2)*a(n-2)) for n > 1.
%F A387308 a(n) = Sum_{k=0..floor(n/2)} 4^k * 5^(n-2*k) * binomial(n+3,n-2*k) * binomial(2*k+3,k).
%F A387308 a(n) = [x^n] (1+5*x+4*x^2)^(n+3).
%F A387308 E.g.f.: exp(5*x) * BesselI(3, 4*x) / 8, with offset 3.
%t A387308 Table[Sum[2^k*Binomial[n+3,k+3]*Binomial[2*k+6,k+6],{k,0,n}],{n,0,25}] (* _Vincenzo Librandi_, Aug 30 2025 *)
%o A387308 (PARI) a(n) = sum(k=0, n, 2^k*binomial(n+3, k+3)*binomial(2*k+6, k+6));
%o A387308 (Magma) [&+[2^k * Binomial(n+3,k+3) * Binomial(2*k+6,k+6): k in [0..n]]: n in [0..25]]; // _Vincenzo Librandi_, Aug 30 2025
%Y A387308 Cf. A331793, A387307.
%K A387308 nonn,new
%O A387308 0,2
%A A387308 _Seiichi Manyama_, Aug 25 2025