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A387275 a(n) = Sum_{k=0..n} 3^(n-k) * binomial(n+2,k+2) * binomial(2*k+4,k+4).

%I A387275 #17 Aug 31 2025 04:37:37
%S A387275 1,15,154,1350,10890,83650,623056,4547520,32735085,233369675,
%T A387275 1652203542,11638252730,81674873553,571575363975,3991529920440,
%U A387275 27829484027400,193791573179883,1348196149698885,9372495529924710,65120144658997050,452263192928596896
%N A387275 a(n) = Sum_{k=0..n} 3^(n-k) * binomial(n+2,k+2) * binomial(2*k+4,k+4).
%H A387275 Vincenzo Librandi, <a href="/A387275/b387275.txt">Table of n, a(n) for n = 0..800</a>
%F A387275 n*(n+4)*a(n) = (n+2) * (5*(2*n+3)*a(n-1) - 21*(n+1)*a(n-2)) for n > 1.
%F A387275 a(n) = Sum_{k=0..floor(n/2)} 5^(n-2*k) * binomial(n+2,n-2*k) * binomial(2*k+2,k).
%F A387275 a(n) = [x^n] (1+5*x+x^2)^(n+2).
%F A387275 E.g.f.: exp(5*x) * BesselI(2, 2*x), with offset 2.
%F A387275 a(n) ~ 7^(n + 5/2) / (2*sqrt(Pi*n)). - _Vaclav Kotesovec_, Aug 31 2025
%t A387275 Table[Sum[3^(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 A387275 (PARI) a(n) = sum(k=0, n, 3^(n-k)*binomial(n+2, k+2)*binomial(2*k+4, k+4));
%o A387275 (Magma) [&+[3^(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 A387275 Cf. A387276, A387277, A387278.
%K A387275 nonn,new
%O A387275 0,2
%A A387275 _Seiichi Manyama_, Aug 24 2025