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%I #33 Sep 14 2021 21:00:55
%S 1,1,1,1,1,1,1,2,1,1,1,6,6,1,1,1,24,90,20,1,1,1,120,2520,1680,70,1,1,
%T 1,720,113400,369600,34650,252,1,1,1,5040,7484400,168168000,63063000,
%U 756756,924,1,1,1,40320,681080400,137225088000,305540235000,11732745024,17153136,3432,1,1
%N Table T(n,k), 0<=k, 0<=n, read by antidiagonals, defined by T(n,k) = (k*n)! / (n!)^k.
%C T(n,k) is the number of lattice paths from {n}^k to {0}^k using steps that decrement one component by 1. - _Alois P. Heinz_, May 06 2013
%H Alois P. Heinz, <a href="/A089759/b089759.txt">Antidiagonals n = 0..32, flattened</a>
%H T. Chappell, A. Lascoux, S. Ole Warnaar, W. Zudilin, <a href="http://arxiv.org/abs/1112.3130">Logarithmic and complex constant term identities</a>, arXiv:1112.3130 [math.CO], 2012.
%e Row n=0: 1, 1, 1, 1, 1, 1, ... A000012
%e Row n=1: 1, 1, 2, 6, 24, 120, ... A000142
%e Row n=2: 1, 1, 6, 90, 2520, 113400, ... A000680
%e Row n=3: 1, 1, 20, 1680, 369600, 168168000, ... A014606
%e Row n=4: 1, 1, 70, 34650, 63063000, 305540235000, ... A014608
%e Row n=5: 1, 1, 252, 756756, 11732745024, 623360743125120, ... A014609
%p T:= (n, k)-> (k*n)!/(n!)^k:
%p seq(seq(T(n, d-n), n=0..d), d=0..10); # _Alois P. Heinz_, Aug 16 2012
%t T[n_, k_] := (k*n)!/(n!)^k; Table[T[n-k, k], {n, 0, 10}, {k, n, 0, -1}] // Flatten (* _Jean-François Alcover_, Dec 19 2015 *)
%Y Cf. A000680, A014606, A014608, A014609, A000984, A187783 (transposed version).
%Y Main diagonal gives A034841.
%Y Cf. A210472, A225094.
%K easy,tabl,nonn
%O 0,8
%A _Philippe Deléham_, Jan 08 2004; revised Jun 08 2005
%E Corrected by _Alois P. Heinz_, Aug 16 2012
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