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A387369 a(n) = Sum_{k=0..n} 2^k * 3^(n-k) * binomial(n+2,k) * binomial(n+2,n-k).

%I A387369 #17 Aug 29 2025 11:28:28
%S A387369 1,15,174,1850,18915,189525,1877596,18476820,181083285,1770245675,
%T A387369 17278828842,168496597230,1642259489143,16002398658225,
%U A387369 155919866646840,1519307275471400,14806582620440553,144329229195062535,1407215890063071910,13724133021646678050,133885448856624266571
%N A387369 a(n) = Sum_{k=0..n} 2^k * 3^(n-k) * binomial(n+2,k) * binomial(n+2,n-k).
%H A387369 Vincenzo Librandi, <a href="/A387369/b387369.txt">Table of n, a(n) for n = 0..800</a>
%F A387369 a(n) = Sum_{k=0..n} 3^k * 2^(n-k) * binomial(n+2,k) * binomial(n+2,n-k).
%F A387369 n*(n+4)*a(n) = (n+2) * (5*(2*n+3)*a(n-1) - (n+1)*a(n-2)) for n > 1.
%F A387369 a(n) = Sum_{k=0..floor(n/2)} 6^k * 5^(n-2*k) * binomial(n+2,n-2*k) * binomial(2*k+2,k).
%F A387369 a(n) = [x^n] (1+5*x+6*x^2)^(n+2).
%F A387369 E.g.f.: exp(5*x) * BesselI(2, 2*sqrt(6)*x) / 6, with offset 2.
%t A387369 Table[Sum[2^k * 3^(n-k)*Binomial[n+2,k]*Binomial[n+2, n-k],{k,0,n}],{n,0,25}] (* _Vincenzo Librandi_, Aug 29 2025 *)
%o A387369 (PARI) a(n) = sum(k=0, n, 2^k*3^(n-k)*binomial(n+2, k)*binomial(n+2, n-k));
%o A387369 (Magma) [&+[2^k * 3^(n-k) * Binomial(n+2,k) * Binomial(n+2,n-k): k in [0..n]]: n in [0..25]]; // _Vincenzo Librandi_, Aug 29 2025
%Y A387369 Cf. A006442, A387368.
%Y A387369 Cf. A387339.
%K A387369 nonn,new
%O A387369 0,2
%A A387369 _Seiichi Manyama_, Aug 27 2025