A360766 a(0) = 0; a(n) = ( (n + sqrt(n))^n - (n - sqrt(n))^n )/(2 * sqrt(n)).

0, 1, 4, 30, 320, 4400, 73872, 1462552, 33325056, 858283776, 24641000000, 779935205984, 26972930949120, 1011642325897216, 40890444454377728, 1771640957790000000, 81896889467638120448, 4022826671022707900416, 209224123984489179202560

Offset: 0

Author

Seiichi Manyama, Mar 11 2023

Keywords

• nonn
• easy

Crossrefs

Main diagonal of A361290.

Cf. A007070, A030192, A093145, A154237.

Cf. A084062.

Programs

PARI

a(n) = polcoeff(lift(Mod('x, ('x-n)^2-n)^n), 1); \\ Kevin Ryde, Mar 16 2023

Formula

a(n) = Sum_{k=0..floor((n-1)/2)} n^(n-1-k) * binomial(n,2*k+1).

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

a(n) = n! * [x^n] exp(n * x) * sinh(sqrt(n) * x) / sqrt(n) for n > 0.