A344639 Array read by ascending antidiagonals: A(n, k) is the number of (n, k)-poly-Cauchy permutations.
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, 97, 32, 1, 5040, 2484, 1392, 854, 531, 275, 64, 1, 40320, 18108, 9260, 5248, 3148, 1817, 793, 128, 1, 362880, 149904, 70508, 36966, 20940, 12134, 6411, 2315, 256, 1
Offset: 0
Examples
n\k| 0 1 2 3 4 ... ---+---------------------------- 0 | 1 1 1 1 1 ... 1 | 1 2 4 8 16 ... 2 | 2 5 13 35 97 ... 3 | 6 17 51 161 531 ... 4 | 24 74 244 854 3148 ... ...
Links
- Beáta Bényi and José Luis Ramírez, Poly-Cauchy numbers -- the combinatorics behind, arXiv:2105.04791 [math.CO], 2021.
Crossrefs
Programs
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Mathematica
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}]]
Formula
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).
A(n, 0) = n! = A000142(n) (see Example 6 in Bényi and Ramírez).
A(1, k) = 2^k = A000079(k) (see Example 7 in Bényi and Ramírez).
A(2, k) = 2^k + 3^k = A007689(k) (see Example 8 in Bényi and Ramírez).
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).
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).
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).
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).
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).
From Seiichi Manyama, Apr 15 2025: (Start)
E.g.f. of column k: Sum_{j>=0} (j+1)^k * (-log(1-x))^j / j!.
E.g.f. of column k: (1/(1-x)) * Sum_{j=0..k} Stirling2(k+1,j+1) * (-log(1-x))^j. (End)
Comments