A357739 - OEIS

%I #10 Oct 12 2022 08:57:32

%S 0,1,1,-2,-23,-99,1,4411,45137,205570,-1270799,-38876573,-441073511,

%T -1921300835,34908578433,994442615986,13032718992033,59450652771077,

%U -1794250960044623,-57608157168424497,-901446808420344919,-5274602459214885362,160827105304127790529

%N a(n) = Sum_{k=0..floor((n-1)/2)} (-n)^k * Stirling2(n,2*k+1).

%H Eric Weisstein's MathWorld, <a href="http://mathworld.wolfram.com/BellPolynomial.html">Bell Polynomial</a>.

%F For n > 0, a(n) = n! * [x^n] sin( sqrt(n) * (exp(x) - 1) )/sqrt(n).

%F For n > 0, a(n) = ( Bell_n(sqrt(n) * i) - Bell_n(-sqrt(n) * i) )/(2 * sqrt(n) * i), where Bell_n(x) is n-th Bell polynomial and i is the imaginary unit.

%o (PARI) a(n) = sum(k=0, (n-1)\2, (-n)^k*stirling(n, 2*k+1, 2));

%o (PARI) a(n) = if(n==0, 0, round(n!*polcoef(sin(sqrt(n)*(exp(x+x*O(x^n))-1))/sqrt(n), n)));

%o (PARI) Bell_poly(n, x) = exp(-x)*suminf(k=0, k^n*x^k/k!);

%o a(n) = if(n==0, 0, round((Bell_poly(n, sqrt(n)*I)-Bell_poly(n, -sqrt(n)*I))/(2*sqrt(n)*I)));

%K sign

%O 0,4

%A _Seiichi Manyama_, Oct 11 2022