This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A387273 #17 Aug 31 2025 10:46:48
%S A387273 1,16,165,1400,10661,75936,517524,3420960,22123530,140782048,
%T A387273 885008839,5511579528,34073731965,209428887360,1281220578936,
%U A387273 7808422173120,47440778110398,287490594872160,1738463164498410,10493677382085744,63245915436539682,380697445274657984
%N A387273 a(n) = Sum_{k=0..n} 2^(n-k) * binomial(n+3,k+3) * binomial(2*k+6,k+6).
%H A387273 Vincenzo Librandi, <a href="/A387273/b387273.txt">Table of n, a(n) for n = 0..800</a>
%F A387273 n*(n+6)*a(n) = (n+3) * (4*(2*n+5)*a(n-1) - 12*(n+2)*a(n-2)) for n > 1.
%F A387273 a(n) = Sum_{k=0..floor(n/2)} 4^(n-2*k) * binomial(n+3,n-2*k) * binomial(2*k+3,k).
%F A387273 a(n) = [x^n] (1+4*x+x^2)^(n+3).
%F A387273 E.g.f.: exp(4*x) * BesselI(3, 2*x), with offset 3.
%t A387273 Table[Sum[2^(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 A387273 (PARI) a(n) = sum(k=0, n, 2^(n-k)*binomial(n+3, k+3)*binomial(2*k+6, k+6));
%o A387273 (Magma) [&+[2^(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 A387273 Cf. A344055, A387272, A387274.
%Y A387273 Cf. A387239, A387276.
%K A387273 nonn,new
%O A387273 0,2
%A A387273 _Seiichi Manyama_, Aug 24 2025