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A387307 a(n) = Sum_{k=0..n} 2^k * binomial(n+2,k+2) * binomial(2*k+4,k+4).

%I A387307 #18 Aug 30 2025 15:56:38
%S A387307 1,15,166,1650,15615,144025,1309084,11793780,105663885,943298675,
%T A387307 8401596258,74716379270,663813240363,5894026429725,52314876771960,
%U A387307 464261939106600,4119843554861913,36560929542771735,324489293583792990,2880380080564191450,25572856871556696471
%N A387307 a(n) = Sum_{k=0..n} 2^k * binomial(n+2,k+2) * binomial(2*k+4,k+4).
%H A387307 Vincenzo Librandi, <a href="/A387307/b387307.txt">Table of n, a(n) for n = 0..800</a>
%F A387307 n*(n+4)*a(n) = (n+2) * (5*(2*n+3)*a(n-1) - 9*(n+1)*a(n-2)) for n > 1.
%F A387307 a(n) = Sum_{k=0..floor(n/2)} 4^k * 5^(n-2*k) * binomial(n+2,n-2*k) * binomial(2*k+2,k).
%F A387307 a(n) = [x^n] (1+5*x+4*x^2)^(n+2).
%F A387307 E.g.f.: exp(5*x) * BesselI(2, 4*x) / 4, with offset 2.
%t A387307 Table[Sum[2^k*Binomial[n+2,k+2]*Binomial[2*k+4,k+4],{k,0,n}],{n,0,25}] (* _Vincenzo Librandi_, Aug 30 2025 *)
%o A387307 (PARI) a(n) = sum(k=0, n, 2^k*binomial(n+2, k+2)*binomial(2*k+4, k+4));
%o A387307 (Magma) [&+[2^k * Binomial(n+2,k+2) * Binomial(2*k+4,k+4): k in [0..n]]: n in [0..25]]; // _Vincenzo Librandi_, Aug 30 2025
%Y A387307 Cf. A331793, A387308.
%K A387307 nonn,new
%O A387307 0,2
%A A387307 _Seiichi Manyama_, Aug 25 2025