A357683 a(n) = Sum_{k=0..floor(n/2)} n^k * |Stirling1(n,2*k)|.

1, 0, 2, 9, 60, 500, 4920, 55566, 706720, 9979200, 154706760, 2609691700, 47547916416, 929943488448, 19421810408000, 431196538865400, 10137091700736000, 251485260368396288, 6563768030597826720, 179746132716715050000, 5152012082327932518400

Offset

0,3

Links

Table of n, a(n) for n=0..20.

Eric Weisstein's World of Mathematics, Pochhammer Symbol.

Formula

a(n) = n! * [x^n] cosh( sqrt(n) * log(1-x) ).

a(n) = ( (sqrt(n))_n + (-sqrt(n))_n )/2, where (x)_n is the Pochhammer symbol.

a(n) ~ n^(n + sqrt(n)/2 - 1/4) / (2*exp(n - sqrt(n) - 1/2)) * (1 - 3/(4*sqrt(n))). - Vaclav Kotesovec, Oct 10 2022

Prog

(PARI) a(n) = sum(k=0, n\2, n^k*abs(stirling(n, 2*k, 1)));
(PARI) a(n) = round(n!*polcoef(cosh(sqrt(n)*log(1-x+x*O(x^n))), n));
(PARI) a(n) = round((prod(k=0, n-1, sqrt(n)+k)+prod(k=0, n-1, -sqrt(n)+k)))/2;

Crossrefs

Cf. A105752, A263687.

Sequence in context: A116364 A354314 A354496 * A392953 A120970 A339360

Adjacent sequences: A357680 A357681 A357682 * A357684 A357685 A357686

Keyword

nonn

Author

Seiichi Manyama, Oct 09 2022

Status

approved