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A387340 a(n) = Sum_{k=0..n} 3^k * binomial(n+3,k) * binomial(n+3,k+3).

%I A387340 #18 Aug 29 2025 11:51:27
%S A387340 1,16,175,1640,14189,117152,939036,7379040,57188010,438810592,
%T A387340 3342302821,25316084248,190937278805,1435287936320,10760879892008,
%U A387340 80509920297792,601343784616830,4485466826475360,33420579148668670,248788060638391120,1850652536242372786
%N A387340 a(n) = Sum_{k=0..n} 3^k * binomial(n+3,k) * binomial(n+3,k+3).
%H A387340 Vincenzo Librandi, <a href="/A387340/b387340.txt">Table of n, a(n) for n = 0..800</a>
%F A387340 n*(n+6)*a(n) = (n+3) * (4*(2*n+5)*a(n-1) - 4*(n+2)*a(n-2)) for n > 1.
%F A387340 a(n) = Sum_{k=0..floor(n/2)} 3^k * 4^(n-2*k) * binomial(n+3,n-2*k) * binomial(2*k+3,k).
%F A387340 a(n) = [x^n] (1+4*x+3*x^2)^(n+3).
%F A387340 E.g.f.: exp(4*x) * BesselI(3, 2*sqrt(3)*x) / (3*sqrt(3)), with offset 3.
%t A387340 Table[Sum[3^k * Binomial[n+3,k]*Binomial[n+3, k+3],{k,0,n}],{n,0,25}] (* _Vincenzo Librandi_, Aug 29 2025 *)
%o A387340 (PARI) a(n) = sum(k=0, n, 3^k*binomial(n+3, k)*binomial(n+3, k+3));
%o A387340 (Magma) [&+[3^k * Binomial(n+3,k) * Binomial(n+3,k+3): k in [0..n]]: n in [0..25]]; // _Vincenzo Librandi_, Aug 29 2025
%Y A387340 Cf. A069835, A331792, A387339.
%Y A387340 Cf. A002696, A387338.
%K A387340 nonn,new
%O A387340 0,2
%A A387340 _Seiichi Manyama_, Aug 27 2025