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%I #23 Jul 30 2020 00:37:09
%S 1,1,-1,1,0,1,1,1,1,-1,1,2,5,2,1,1,3,13,29,9,-1,1,4,25,116,233,44,1,1,
%T 5,41,299,1393,2329,265,-1,1,6,61,614,4785,20894,27949,1854,1,1,7,85,
%U 1097,12281,95699,376093,391285,14833,-1,1,8,113,1784,26329,307024,2296777,7897952,6260561,133496,1
%N Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of the e.g.f. exp(-x)/(1 - k*x).
%C For n > 0 and k > 0, A(n,k) gives the number of derangements of the generalized symmetric group S(k,n), which is the wreath product of Z_k by S_n. - _Peter Kagey_, Apr 07 2020
%H Alois P. Heinz, <a href="/A320032/b320032.txt">Antidiagonals n = 0..140, flattened</a>
%H Sami H. Assaf, <a href="https://arxiv.org/abs/1002.3138">Cyclic derangements</a>, arXiv:1002.3138 [math.CO], 2010.
%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Generalized_symmetric_group">Generalized symmetric group</a>.
%F E.g.f. of column k: exp(-x)/(1 - k*x).
%F A(n,k) = Sum_{j=0..n} (-1)^(n-j)*binomial(n,j)*j!*k^j.
%F A(n,k) = (-1)^n*2F0(1,-n; ; k).
%e E.g.f. of column k: A_k(x) = 1 + (k - 1)*x/1! + (2*k^2 - 2*k + 1)*x^2/2! + (6*k^3 - 6*k^2 + 3*k - 1)*x^3/3! + (24*k^4 - 24*k^3 + 12*k^2 - 4*k + 1)*x^4/4! + ...
%e Square array begins:
%e 1, 1, 1, 1, 1, 1, ...
%e -1, 0, 1, 2, 3, 4, ...
%e 1, 1, 5, 13, 25, 41, ...
%e -1, 2, 29, 116, 299, 614, ...
%e 1, 9, 233, 1393, 4785, 12281, ...
%e -1, 44, 2329, 20894, 95699, 307024, ...
%p A:= proc(n, k) option remember;
%p `if`(n=0, 1, k*n*A(n-1, k)+(-1)^n)
%p end:
%p seq(seq(A(n, d-n), n=0..d), d=0..12); # _Alois P. Heinz_, May 07 2020
%t Table[Function[k, n! SeriesCoefficient[Exp[-x]/(1 - k x), {x, 0, n}]][j - n], {j, 0, 10}, {n, 0, j}] // Flatten
%t Table[Function[k, (-1)^n HypergeometricPFQ[{1, -n}, {}, k]][j - n], {j, 0, 10}, {n, 0, j}] // Flatten
%Y Columns k=0..5 give A033999, A000166, A000354, A000180, A001907, A001908.
%Y Main diagonal gives A319392.
%Y Cf. A320031.
%K sign,tabl
%O 0,12
%A _Ilya Gutkovskiy_, Oct 03 2018