A357721 a(n) = Sum_{k=0..floor(n/2)} (-n)^k * Stirling1(n,2*k).

1, 0, -2, 9, -28, 0, 1200, -16464, 167904, -1393200, 7429240, 43124400, -2404571904, 55590286752, -1027511503200, 16489054310400, -222885864448000, 1994839594780032, 14489184835474272, -1470395490046560000, 54581408106475622400, -1608207353670788640000

Offset

0,3

Links

Table of n, a(n) for n=0..21.

Eric Weisstein's World of Mathematics, Pochhammer Symbol.

Formula

a(n) = n! * [x^n] cos( sqrt(n) * log(1+x) ).

a(n) = (-1)^n * ( (sqrt(n) * i)_n + (-sqrt(n) * i)_n )/2, where (x)_n is the Pochhammer symbol and i is the imaginary unit.

Prog

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

Crossrefs

Main diagonal of A357720.

Cf. A357683, A357729.

Sequence in context: A115186 A243560 A090900 * A155472 A100293 A202679

Adjacent sequences: A357718 A357719 A357720 * A357722 A357723 A357724

Keyword

sign

Author

Seiichi Manyama, Oct 10 2022

Status

approved