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

0,2

Comments

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

Links

Vincenzo Librandi, Table of n, a(n) for n = 0..300

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

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 *)

Prog

(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));
(Magma) [&+[n^(n-2*k) *Binomial(2*(n-k), n-k) * Binomial(n-k, k): k in [0..Floor (n/2)]]: n in [0..35]]; // Vincenzo Librandi, Sep 04 2025

Crossrefs

Main diagonal of A386621.

Cf. A110129, A387459.

Sequence in context: A379247 A379257 A255538 * A302719 A377527 A377547

Adjacent sequences: A387427 A387428 A387429 * A387431 A387432 A387433

Keyword

nonn

Author

Seiichi Manyama, Aug 29 2025

Status

approved