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%I #20 May 26 2021 02:24:07
%S 1,1,1,2,2,1,6,5,4,1,24,17,13,8,1,120,74,51,35,16,1,720,394,244,161,
%T 97,32,1,5040,2484,1392,854,531,275,64,1,40320,18108,9260,5248,3148,
%U 1817,793,128,1,362880,149904,70508,36966,20940,12134,6411,2315,256,1
%N Array read by ascending antidiagonals: A(n, k) is the number of (n, k)-poly-Cauchy permutations.
%C An (n, k)-poly-Cauchy permutation is a permutation which satisfies the properties listed by Bényi and Ramírez in Definition 1.
%H Beáta Bényi and José Luis Ramírez, <a href="https://arxiv.org/abs/2105.04791">Poly-Cauchy numbers -- the combinatorics behind</a>, arXiv:2105.04791 [math.CO], 2021.
%F A(n, k) = Sum_{m=0..n} abs(S1(n, m))*(m + 1)^k, where S1 indicates the signed Stirling numbers of first kind (see Theorem 5 in Bényi and Ramírez).
%F A(n, 0) = n! = A000142(n) (see Example 6 in Bényi and Ramírez).
%F A(1, k) = 2^k = A000079(k) (see Example 7 in Bényi and Ramírez).
%F A(2, k) = 2^k + 3^k = A007689(k) (see Example 8 in Bényi and Ramírez).
%F Sum_{m=0..n} (-1)^m*S2(n, m)*A(m, k) = (-1)^n*(n + 1)^k, where S2 indicates the Stirling numbers of the second kind (see Theorem 9 in Bényi and Ramírez).
%F A(n, k) = Sum_{j=0..k} j!*abs(S1(n+1, j+1))*S2(k+1, j+1) (see Theorem 14 in Bényi and Ramírez).
%F A(n, k) = (n - 1)*A(n-1, k) + Sum_{i=0..k} C(k, i)*A(n-1, k-i) for n > 0 (see Theorem 15 in Bényi and Ramírez).
%F A(n, k) = Sum_{i=0..n} Sum_{j=0..k} C(n-1, i)*i!*C(k, j)*A(n-1-i, k-j) for n > 0 (see Theorem 17 in Bényi and Ramírez).
%F A(n, k) = Sum_{m=0..n} Sum_{i=0..m} C(k-i, m-i)*S2(k, i)*abs(S1(n+1, m+1)) (see Theorem 18 in Bényi and Ramírez).
%e n\k| 0 1 2 3 4 ...
%e ---+----------------------------
%e 0 | 1 1 1 1 1 ...
%e 1 | 1 2 4 8 16 ...
%e 2 | 2 5 13 35 97 ...
%e 3 | 6 17 51 161 531 ...
%e 4 | 24 74 244 854 3148 ...
%e ...
%t A[n_,k_]:=Sum[Abs[StirlingS1[n,m]](m+1)^k,{m,0,n}]; Flatten[Table[A[n-k,k],{n,0,9},{k,0,n}]]
%Y Cf. A000012 (n = 0), A000079 (n = 1), A000142 (k = 0), A000774 (k = 1), A007318, A007689 (n = 2), A008275, A008277, A081048, A192563 (diagonal), A223899 (k = 2), A223901 (k = 3), A223902 (k = 4), A223904 (k = 5), A344640 (antidiagonal sums).
%K nonn,tabl
%O 0,4
%A _Stefano Spezia_, May 25 2021