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 A387272 #16 Aug 31 2025 12:51:30
%S A387272 1,12,100,720,4815,30884,193144,1188576,7236690,43741720,263056728,
%T A387272 1576298464,9421080123,56200937940,334801389360,1992471776448,
%U A387272 11848869296622,70425535830696,418426332826200,2485390365370080,14760336569524854,87650482093915752
%N A387272 a(n) = Sum_{k=0..n} 2^(n-k) * binomial(n+2,k+2) * binomial(2*k+4,k+4).
%H A387272 Vincenzo Librandi, <a href="/A387272/b387272.txt">Table of n, a(n) for n = 0..800</a>
%F A387272 n*(n+4)*a(n) = (n+2) * (4*(2*n+3)*a(n-1) - 12*(n+1)*a(n-2)) for n > 1.
%F A387272 a(n) = Sum_{k=0..floor(n/2)} 4^(n-2*k) * binomial(n+2,n-2*k) * binomial(2*k+2,k).
%F A387272 a(n) = [x^n] (1+4*x+x^2)^(n+2).
%F A387272 E.g.f.: exp(4*x) * BesselI(2, 2*x), with offset 2.
%t A387272 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 *)
%o A387272 (PARI) a(n) = sum(k=0, n, 2^(n-k)*binomial(n+2, k+2)*binomial(2*k+4, k+4));
%o A387272 (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
%Y A387272 Cf. A344055, A387273, A387274.
%Y A387272 Cf. A387280.
%K A387272 nonn,new
%O A387272 0,2
%A A387272 _Seiichi Manyama_, Aug 24 2025