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A357729 a(n) = Sum_{k=0..floor(n/2)} (-n)^k * Stirling2(n,2*k). 3
1, 0, -2, -9, -12, 175, 1938, 9506, -24248, -1065663, -12021610, -56195425, 677072220, 19979234080, 251733387514, 1135594212255, -29317384858352, -901607623649489, -13233854770928514, -68574233644270566, 2258648937829442660, 81748108921355457777 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,3
LINKS
Eric Weisstein's World of Mathematics, Bell Polynomial.
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.
PROG
(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;
CROSSREFS
Main diagonal of A357728.
Sequence in context: A225548 A233400 A234945 * A271646 A024976 A171234
KEYWORD
sign
AUTHOR
Seiichi Manyama, Oct 11 2022
STATUS
approved

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Last modified February 29 04:26 EST 2024. Contains 370401 sequences. (Running on oeis4.)