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A306641 A(n,k) = Sum_{j=0..n} (k*n)!/(j! * (n-j)!)^k, square array A(n,k) read by antidiagonals, for n >= 0, k >= 0.

%I A306641 #30 Feb 16 2023 05:25:07
%S A306641 1,1,2,1,2,3,1,4,4,4,1,12,36,8,5,1,48,900,400,16,6,1,240,45360,94080,
%T A306641 4900,32,7,1,1440,3855600,60614400,11988900,63504,64,8,1,10080,
%U A306641 493970400,82065984000,114144030000,1704214512,853776,128,9
%N A306641 A(n,k) = Sum_{j=0..n} (k*n)!/(j! * (n-j)!)^k, square array A(n,k) read by antidiagonals, for n >= 0, k >= 0.
%C A306641 Columns are the number of maximal chains to multiples of J in the graded poset of (2 X n) antimagic squares. See Stanley. - _Arnav Krishna_, Jan 13 2023
%D A306641 R. P. Stanley, Enumerative Combinatorics, Vol I, Exercise 53, p. 540.
%H A306641 Seiichi Manyama, <a href="/A306641/b306641.txt">Antidiagonals n = 0..49, flattened</a>
%e A306641 Square array begins:
%e A306641    1,  1,     1,          1,               1, ...
%e A306641    2,  2,     4,         12,              48, ...
%e A306641    3,  4,    36,        900,           45360, ...
%e A306641    4,  8,   400,      94080,        60614400, ...
%e A306641    5, 16,  4900,   11988900,    114144030000, ...
%e A306641    6, 32, 63504, 1704214512, 249344297250048, ...
%Y A306641 Columns 0-3 give A000027(n+1), A000079, A002894, A306642, A345646.
%Y A306641 Rows 0-1 give A000012, A208529(n+2).
%Y A306641 Main diagonal gives A306644.
%K A306641 nonn,tabl
%O A306641 0,3
%A A306641 _Seiichi Manyama_, Mar 02 2019