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 A387339 #18 Aug 29 2025 14:34:35
%S A387339 1,12,108,880,6855,52164,391720,2918304,21634290,159880600,1179180552,
%T A387339 8685874080,63930198787,470327654580,3459353475600,25442360389696,
%U A387339 187126561024686,1376455855989672,10126540146288520,74515694338112160,548444877468906726
%N A387339 a(n) = Sum_{k=0..n} 3^k * binomial(n+2,k) * binomial(n+2,k+2).
%H A387339 Vincenzo Librandi, <a href="/A387339/b387339.txt">Table of n, a(n) for n = 0..800</a>
%F A387339 n*(n+4)*a(n) = (n+2) * (4*(2*n+3)*a(n-1) - 4*(n+1)*a(n-2)) for n > 1.
%F A387339 a(n) = Sum_{k=0..floor(n/2)} 3^k * 4^(n-2*k) * binomial(n+2,n-2*k) * binomial(2*k+2,k).
%F A387339 a(n) = [x^n] (1+4*x+3*x^2)^(n+2).
%F A387339 E.g.f.: exp(4*x) * BesselI(2, 2*sqrt(3)*x) / 3, with offset 2.
%t A387339 Table[Sum[3^k * Binomial[n+2,k]*Binomial[n+2, k+2],{k,0,n}],{n,0,25}] (* _Vincenzo Librandi_, Aug 29 2025 *)
%o A387339 (PARI) a(n) = sum(k=0, n, 3^k*binomial(n+2, k)*binomial(n+2, k+2));
%o A387339 (Magma) [&+[3^k * Binomial(n+2,k) * Binomial(n+2,k+2): k in [0..n]]: n in [0..25]]; // _Vincenzo Librandi_, Aug 29 2025
%Y A387339 Cf. A069835, A331792, A387340.
%Y A387339 Cf. A387310.
%K A387339 nonn,new
%O A387339 0,2
%A A387339 _Seiichi Manyama_, Aug 27 2025