cp's OEIS Frontend

A320032 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).

%I A320032 #29 Jun 06 2025 08:35:26
%S A320032 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 A320032 5,41,299,1393,2329,265,-1,1,6,61,614,4785,20894,27949,1854,1,1,7,85,
%U A320032 1097,12281,95699,376093,391285,14833,-1,1,8,113,1784,26329,307024,2296777,7897952,6260561,133496,1
%N A320032 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 A320032 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 A320032 Alois P. Heinz, <a href="/A320032/b320032.txt">Antidiagonals n = 0..140, flattened</a>
%H A320032 Sami H. Assaf, <a href="https://arxiv.org/abs/1002.3138">Cyclic derangements</a>,  arXiv:1002.3138 [math.CO], 2010.
%H A320032 Hilarion L. M. Faliharimalala and Jiang Zeng, <a href="https://doi.org/10.37236/789">Derangements and Euler's difference table for C_l wr S_n</a>, Electron. J. Combin. 15 (2008), #R65.
%H A320032 Wikipedia, <a href="https://en.wikipedia.org/wiki/Generalized_symmetric_group">Generalized symmetric group</a>.
%F A320032 E.g.f. of column k: exp(-x)/(1 - k*x).
%F A320032 A(n,k) = Sum_{j=0..n} (-1)^(n-j)*binomial(n,j)*j!*k^j.
%F A320032 A(n,k) = (-1)^n*2F0(1,-n; ; k).
%e A320032 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 A320032 Square array begins:
%e A320032    1,   1,     1,      1,      1,       1,  ...
%e A320032   -1,   0,     1,      2,      3,       4,  ...
%e A320032    1,   1,     5,     13,     25,      41,  ...
%e A320032   -1,   2,    29,    116,    299,     614,  ...
%e A320032    1,   9,   233,   1393,   4785,   12281,  ...
%e A320032   -1,  44,  2329,  20894,  95699,  307024,  ...
%p A320032 A:= proc(n, k) option remember;
%p A320032      `if`(n=0, 1, k*n*A(n-1, k)+(-1)^n)
%p A320032     end:
%p A320032 seq(seq(A(n, d-n), n=0..d), d=0..12);  # _Alois P. Heinz_, May 07 2020
%t A320032 Table[Function[k, n! SeriesCoefficient[Exp[-x]/(1 - k x), {x, 0, n}]][j - n], {j, 0, 10}, {n, 0, j}] // Flatten
%t A320032 Table[Function[k, (-1)^n HypergeometricPFQ[{1, -n}, {}, k]][j - n], {j, 0, 10}, {n, 0, j}] // Flatten
%Y A320032 Columns k=0..5 give A033999, A000166, A000354, A000180, A001907, A001908.
%Y A320032 Main diagonal gives A319392.
%Y A320032 Cf. A320031.
%K A320032 sign,tabl
%O A320032 0,12
%A A320032 _Ilya Gutkovskiy_, Oct 03 2018