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

1, 2, 26, 576, 18886, 822800, 44758244, 2920443904, 222277449286, 19333107926208, 1891679562586252, 205658657276205056, 24594577004735218716, 3208651043895419972096, 453493188773477070618248, 69025100503218462336614400, 11256667883184684951198851654, 1958143582960886584057480612864

Offset: 0

Author

Seiichi Manyama, Aug 29 2025

Keywords

• nonn
• new

Comments

Sum_{k=0..n} (n-i)^k * (n+i)^(n-k) * binomial(n,k) = 2^n * n^n. - Vaclav Kotesovec, Aug 29 2025

Crossrefs

Main diagonal of A386621.

Cf. A110129, A387459.

Programs

Mathematica

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 *)
Table[(I + n)^n Hypergeometric2F1[-n, -n, 1, (-I + n)/(I + n)], {n, 0, 20}] (* Vaclav Kotesovec, Aug 29 2025 *)

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));

Formula

a(n) = Sum_{k=0..floor(n/2)} n^(n-2*k) * binomial(2*(n-k),n-k) * binomial(n-k,k).

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).

a(n) = [x^n] (1 + 2*n*x + (n^2+1)*x^2)^n.

a(n) ~ 2^(2*n) * n^(n - 1/2) / sqrt(Pi). - Vaclav Kotesovec, Aug 29 2025