A357729 a(n) = Sum_{k=0..floor(n/2)} (-n)^k * Stirling2(n,2*k).

1, 0, -2, -9, -12, 175, 1938, 9506, -24248, -1065663, -12021610, -56195425, 677072220, 19979234080, 251733387514, 1135594212255, -29317384858352, -901607623649489, -13233854770928514, -68574233644270566, 2258648937829442660, 81748108921355457777

Offset: 0

Author

Seiichi Manyama, Oct 11 2022

Keywords

• sign

Links

• Andrew Howroyd, Table of n, a(n) for n = 0..200
• Eric Weisstein's World of Mathematics, Bell Polynomial.

Crossrefs

Main diagonal of A357728.

Cf. A357682, A357721.

Programs

PARI

a(n) = sum(k=0, n\2, (-n)^k*stirling(n, 2*k, 2));

PARI

a(n) = round(n!*polcoef(cos(sqrt(n)*(exp(x+x*O(x^n))-1)), n));

PARI

Bell_poly(n, x) = exp(-x)*suminf(k=0, k^n*x^k/k!);
a(n) = round((Bell_poly(n, sqrt(n)*I)+Bell_poly(n, -sqrt(n)*I)))/2;

Formula

a(n) = n! * [x^n] cos( sqrt(n) * (exp(x) - 1) ).

a(n) = ( Bell_n(sqrt(n) * i) + Bell_n(-sqrt(n) * i) )/2, where Bell_n(x) is n-th Bell polynomial and i is the imaginary unit.