%I #13 Sep 01 2024 14:36:09

%S 1,1,0,-9,-40,125,3444,18571,-241872,-5796711,-24387220,1132278191,

%T 25132445832,8850583573,-10681029498972,-214099676807085,

%U 1643397436986464,176719161389104817,2976468247699317468,-71662294521163070153,-4638920054290748840520,-55645074852328083377619

%N a(n) = Sum_{k=0..n} (-1)^(n-k)*binomial(n, k)*Stirling2(n, k)*k!.

%C Inverse binomial convolution of the Fubini numbers (A131689).

%F a(n) = Sum_{k=0..n} (-1)^k * A219859(n,k). - _Alois P. Heinz_, Jan 24 2022

%F a(n) = n! * [x^n] (2 - exp(-x))^n. - _Fabian Pereyra_, Aug 31 2024

%t a[n_] := Sum[(-1)^(n - k) * Binomial[n, k] * StirlingS2[n, k] * k!, {k, 0, n}]; Array[a, 22, 0] (* _Amiram Eldar_, May 10 2021 *)

%o (PARI) a(n) = sum(k=0, n, (-1)^(n-k)*binomial(n, k)*stirling(n, k, 2)*k!); \\ _Michel Marcus_, May 10 2021

%Y Cf. A000312, A131689, A122455, A343841, A317274, A211210, A219859.

%K sign

%O 0,4

%A _Peter Luschny_, May 10 2021