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A387402 a(n) = Sum_{k=0..n} (1-i)^k * (1+i)^(n-k) * binomial(n+2,k) * binomial(n+2,n-k), where i is the imaginary unit.

%I A387402 #17 Sep 04 2025 03:07:16
%S A387402 1,6,32,160,780,3752,17920,85248,404640,1918400,9090048,43064320,
%T A387402 204032192,966887040,4583424000,21735350272,103114538496,489392157696,
%U A387402 2323701678080,11037970513920,52454251902976,249373626208256,1186024281341952,5642924625100800,26858183388774400,127880625111662592
%N A387402 a(n) = Sum_{k=0..n} (1-i)^k * (1+i)^(n-k) * binomial(n+2,k) * binomial(n+2,n-k), where i is the imaginary unit.
%H A387402 Vincenzo Librandi, <a href="/A387402/b387402.txt">Table of n, a(n) for n = 0..1000</a>
%F A387402 n*(n+4)*a(n) = (n+2) * (2*(2*n+3)*a(n-1) + 4*(n+1)*a(n-2)) for n > 1.
%F A387402 a(n) = Sum_{k=0..floor(n/2)} 2^(n-k) * binomial(n+2,n-2*k) * binomial(2*k+2,k).
%F A387402 a(n) = [x^n] (1+2*x+2*x^2)^(n+2).
%F A387402 E.g.f.: exp(2*x) * BesselI(2, 2*sqrt(2)*x) / 2, with offset 2.
%t A387402 Table[Sum[2^(n-k)*Binomial[n+2,n-2*k]*Binomial[2*k+2,k],{k,0,Floor[n/2]}],{n,0,30}] (* _Vincenzo Librandi_, Sep 04 2025 *)
%o A387402 (PARI) a(n) = sum(k=0, n\2, 2^(n-k)*binomial(n+2, n-2*k)*binomial(2*k+2, k));
%o A387402 (Magma) [&+[2^(n-k) * Binomial(n+2,n-2*k) * Binomial(2*k+2,k): k in [0..Floor (n/2)]]: n in [0..35]]; // _Vincenzo Librandi_, Sep 04 2025
%Y A387402 Cf. A006139, A387401, A387403.
%Y A387402 Cf. A374511.
%K A387402 nonn,new
%O A387402 0,2
%A A387402 _Seiichi Manyama_, Aug 29 2025