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A387401 a(n) = Sum_{k=0..n} (1-i)^k * (1+i)^(n-k) * binomial(n+1,k) * binomial(n+1,n-k), where i is the imaginary unit.

%I A387401 #19 Sep 04 2025 03:08:00
%S A387401 1,4,18,80,360,1632,7448,34176,157536,728960,3384128,15754752,
%T A387401 73525504,343870464,1611288960,7562801152,35550504448,167339022336,
%U A387401 788643765248,3720901222400,17573439614976,83074892775424,393056192851968,1861155016212480,8819174122700800,41818448615636992
%N A387401 a(n) = Sum_{k=0..n} (1-i)^k * (1+i)^(n-k) * binomial(n+1,k) * binomial(n+1,n-k), where i is the imaginary unit.
%H A387401 Vincenzo Librandi, <a href="/A387401/b387401.txt">Table of n, a(n) for n = 0..1000</a>
%F A387401 n*(n+2)*a(n) = (n+1) * (2*(2*n+1)*a(n-1) + 4*n*a(n-2)) for n > 1.
%F A387401 a(n) = Sum_{k=0..floor(n/2)} 2^(n-k) * binomial(n+1,n-2*k) * binomial(2*k+1,k).
%F A387401 a(n) = [x^n] (1+2*x+2*x^2)^(n+1).
%F A387401 E.g.f.: exp(2*x) * BesselI(1, 2*sqrt(2)*x) / sqrt(2), with offset 1.
%F A387401 a(n) = (n+1) * A071356(n).
%t A387401 Table[Sum[2^(n-k)*Binomial[n+1,n-2*k]*Binomial[2*k+1,k],{k,0,Floor[n/2]}],{n,0,30}] (* _Vincenzo Librandi_, Sep 04 2025 *)
%o A387401 (PARI) a(n) = sum(k=0, n\2, 2^(n-k)*binomial(n+1, n-2*k)*binomial(2*k+1, k));
%o A387401 (Magma) [&+[2^(n-k) * Binomial(n+1,n-2*k) * Binomial(2*k+1,k): k in [0..Floor (n/2)]]: n in [0..35]]; // _Vincenzo Librandi_, Sep 04 2025
%Y A387401 Cf. A006139, A387402, A387403.
%Y A387401 Cf. A071356, A374497.
%K A387401 nonn,new
%O A387401 0,2
%A A387401 _Seiichi Manyama_, Aug 29 2025