cp's OEIS Frontend

A387403 a(n) = Sum_{k=0..n} (1-i)^k * (1+i)^(n-k) * binomial(n+3,k) * binomial(n+3,n-k), where i is the imaginary unit.

%I A387403 #17 Sep 04 2025 03:28:28
%S A387403 1,8,50,280,1484,7616,38304,190080,934560,4564736,22189024,107476096,
%T A387403 519180480,2502850560,12046666752,57912029184,278136798720,
%U A387403 1334832967680,6402435630080,30695114813440,147110418036736,704860523102208,3376580007936000,16172904859238400
%N A387403 a(n) = Sum_{k=0..n} (1-i)^k * (1+i)^(n-k) * binomial(n+3,k) * binomial(n+3,n-k), where i is the imaginary unit.
%H A387403 Vincenzo Librandi, <a href="/A387403/b387403.txt">Table of n, a(n) for n = 0..1000</a>
%F A387403 n*(n+6)*a(n) = (n+3) * (2*(2*n+5)*a(n-1) + 4*(n+2)*a(n-2)) for n > 1.
%F A387403 a(n) = Sum_{k=0..floor(n/2)} 2^(n-k) * binomial(n+3,n-2*k) * binomial(2*k+3,k).
%F A387403 a(n) = [x^n] (1+2*x+2*x^2)^(n+3).
%F A387403 E.g.f.: exp(2*x) * BesselI(3, 2*sqrt(2)*x) / (2*sqrt(2)), with offset 3.
%t A387403 Table[Sum[2^(n-k)*Binomial[n+3,n-2*k]*Binomial[2*k+3,k],{k,0,Floor[n/2]}],{n,0,30}] (* _Vincenzo Librandi_, Sep 04 2025 *)
%o A387403 (PARI) a(n) = sum(k=0, n\2, 2^(n-k)*binomial(n+3, n-2*k)*binomial(2*k+3, k));
%o A387403 (Magma) [&+[2^(n-k) * Binomial(n+3,n-2*k) * Binomial(2*k+3,k): k in [0..Floor (n/2)]]: n in [0..35]]; // _Vincenzo Librandi_, Sep 04 2025
%Y A387403 Cf. A006139, A387401, A387402.
%Y A387403 Cf. A374513.
%K A387403 nonn,new
%O A387403 0,2
%A A387403 _Seiichi Manyama_, Aug 29 2025