|
%I #30 Jan 08 2022 22:08:36
%S 1,2,2,4,7,4,8,23,23,8,16,73,115,73,16,32,227,533,533,227,32,64,697,
%T 2359,3451,2359,697,64,128,2123,10133,20753,20753,10133,2123,128,256,
%U 6433,42655,118843,164731,118843,42655,6433,256,512,19427,177053,657833,1220657,1220657,657833,177053,19427,512
%N 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.
%C 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.
%H Arvind Ayyer and Beáta Bényi, <a href="https://doi.org/10.37236/10420">Toppling on permutations with an extra chip</a>, El. J. Comb., 28 (2021), P4.18. The array seems to appear in Table 6.
%H Usman Hafeez, Peter Marcus, Kyle Ormsby and Angélica Osorno, <a href="https://arxiv.org/abs/2109.08210">Saturated and linear isometric transfer systems for cyclic groups of order p^m*q^n</a>, arXiv:2109.08210 [math.AT], 2021.
%F T(n,k) = T(k,n).
%F 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
%e Array begins:
%e 1 2 4 8 16 32 ...
%e 2 7 23 73 227 697 ...
%e 4 23 115 533 2359 10133 ...
%e 8 73 533 3451 20753 118843 ...
%e 16 227 2359 20753 164731 1220657 ...
%e 32 697 10133 118843 1220657 11467387 ...
%e ...
%o (PARI) T(n, k) = sum(j=2, n+2, (-1)^(n-j)*stirling(n+1, j-1, 2)*j!*j^k/2);
%Y Columns k=0-1 gives A000079, A083313(n+1).
%Y Main diagonal gives A220181(n+1).
%Y Cf. A008277 (Stirling2), A143494.
%K nonn,tabl
%O 0,2
%A _Michel Marcus_, Sep 20 2021
|
