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 A387278 #17 Aug 30 2025 18:45:12
%S A387278 1,10,78,560,3885,26550,180285,1221400,8272251,56062550,380361212,
%T A387278 2583867720,17575724491,119705522370,816297170310,5572945684800,
%U A387278 38088275031435,260576833989150,1784382167211378,12229792774162800,83888652677196591,575858959975595010
%N A387278 a(n) = Sum_{k=0..n} 3^(n-k) * binomial(n+1,k+1) * binomial(2*k+2,k+2).
%H A387278 Vincenzo Librandi, <a href="/A387278/b387278.txt">Table of n, a(n) for n = 0..800</a>
%F A387278 n*(n+2)*a(n) = (n+1) * (5*(2*n+1)*a(n-1) - 21*n*a(n-2)) for n > 1.
%F A387278 a(n) = Sum_{k=0..floor(n/2)} 5^(n-2*k) * binomial(n+1,n-2*k) * binomial(2*k+1,k).
%F A387278 a(n) = [x^n] (1+5*x+x^2)^(n+1).
%F A387278 E.g.f.: exp(5*x) * BesselI(1, 2*x), with offset 1.
%t A387278 Table[Sum[3^(n-k)*Binomial[n+1,k+1]*Binomial[2*k+2,k+2],{k,0,n}],{n,0,25}] (* _Vincenzo Librandi_, Aug 30 2025 *)
%o A387278 (PARI) a(n) = sum(k=0, n, 3^(n-k)*binomial(n+1, k+1)*binomial(2*k+2, k+2));
%o A387278 (Magma) [&+[3^(n-k) * Binomial(n+1,k+1) * Binomial(2*k+2,k+2): k in [0..n]]: n in [0..25]]; // _Vincenzo Librandi_, Aug 30 2025
%Y A387278 Cf. A098409, A387275, A387276, A387277.
%Y A387278 Cf. A026376, A344055.
%K A387278 nonn,new
%O A387278 0,2
%A A387278 _Seiichi Manyama_, Aug 24 2025