A347940 Array T(n, k) = Sum_{j=2..n+2} (-1)^(n-j)*Stirling2(n+1, j-1)*j!*j^k/2, for n and k >= 0, read by antidiagonals.

1, 2, 2, 4, 7, 4, 8, 23, 23, 8, 16, 73, 115, 73, 16, 32, 227, 533, 533, 227, 32, 64, 697, 2359, 3451, 2359, 697, 64, 128, 2123, 10133, 20753, 20753, 10133, 2123, 128, 256, 6433, 42655, 118843, 164731, 118843, 42655, 6433, 256, 512, 19427, 177053, 657833, 1220657, 1220657, 657833, 177053, 19427, 512

Offset: 0

Author

Michel Marcus, Sep 20 2021

Keywords

• nonn
• tabl

Comments

T(m, n) is the number of saturated Cp^m*q^n-transfer systems where Cp^m*q^n is the cyclic group of order p^m*q^n, for m, n >= 0, p and q primes. See Hafeez et al. link page 1.

Examples

			Array begins:
   1   2     4      8      16       32 ...
   2   7    23     73     227      697 ...
   4  23   115    533    2359    10133 ...
   8  73   533   3451   20753   118843 ...
  16 227  2359  20753  164731  1220657 ...
  32 697 10133 118843 1220657 11467387 ...
  ...
		

Links

• Arvind Ayyer and Beáta Bényi, Toppling on permutations with an extra chip, El. J. Comb., 28 (2021), P4.18. The array seems to appear in Table 6.
• Usman Hafeez, Peter Marcus, Kyle Ormsby and Angélica Osorno, Saturated and linear isometric transfer systems for cyclic groups of order p^m*q^n, arXiv:2109.08210 [math.AT], 2021.

Crossrefs

Columns k=0-1 gives A000079, A083313(n+1).

Main diagonal gives A220181(n+1).

Cf. A008277 (Stirling2), A143494.

Programs

PARI

T(n, k) = sum(j=2, n+2, (-1)^(n-j)*stirling(n+1, j-1, 2)*j!*j^k/2);

Formula

T(n,k) = T(k,n).

T(n,k) = Sum_{j=0..min(n,k)} (j!*(j+2)!/2)*Stirling2(n+2,j+2;2)*Stirling2(k+2,j+2;2), n,k >= 0, where Stirling2(n,k;2) are the 2-Stirling numbers of the second kind A143494. - Fabián Pereyra, Jan 08 2022