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A387430 a(n) = Sum_{k=0..n} (n-i)^k * (n+i)^(n-k) * binomial(n,k)^2, where i is the imaginary unit.

%I A387430 #20 Aug 29 2025 15:05:04
%S A387430 1,2,26,576,18886,822800,44758244,2920443904,222277449286,
%T A387430 19333107926208,1891679562586252,205658657276205056,
%U A387430 24594577004735218716,3208651043895419972096,453493188773477070618248,69025100503218462336614400,11256667883184684951198851654,1958143582960886584057480612864
%N A387430 a(n) = Sum_{k=0..n} (n-i)^k * (n+i)^(n-k) * binomial(n,k)^2, where i is the imaginary unit.
%C A387430 Sum_{k=0..n} (n-i)^k * (n+i)^(n-k) * binomial(n,k) = 2^n * n^n. - _Vaclav Kotesovec_, Aug 29 2025
%F A387430 a(n) = Sum_{k=0..floor(n/2)} n^(n-2*k) * binomial(2*(n-k),n-k) * binomial(n-k,k).
%F A387430 a(n) = Sum_{k=0..floor(n/2)} (n^2+1)^k * (2*n)^(n-2*k) * binomial(n,2*k) * binomial(2*k,k).
%F A387430 a(n) = [x^n] (1 + 2*n*x + (n^2+1)*x^2)^n.
%F A387430 a(n) ~ 2^(2*n) * n^(n - 1/2) / sqrt(Pi). - _Vaclav Kotesovec_, Aug 29 2025
%t A387430 Join[{1}, Table[Sum[(n^2 + 1)^k * (2*n)^(n-2*k) * Binomial[n,2*k] * Binomial[2*k,k], {k,0,n/2}], {n,1,20}]] (* or *)
%t A387430 Table[(I + n)^n Hypergeometric2F1[-n, -n, 1, (-I + n)/(I + n)], {n, 0, 20}] (* _Vaclav Kotesovec_, Aug 29 2025 *)
%o A387430 (PARI) a(n) = sum(k=0, n\2, (n^2+1)^k*(2*n)^(n-2*k)*binomial(n, 2*k)*binomial(2*k, k));
%Y A387430 Main diagonal of A386621.
%Y A387430 Cf. A110129, A387459.
%K A387430 nonn,new
%O A387430 0,2
%A A387430 _Seiichi Manyama_, Aug 29 2025