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%I #23 Jul 16 2022 11:49:11
%S 1,0,3,3,0,6,0,15,0,10,15,0,45,0,15,0,105,0,105,0,21,105,0,420,0,210,
%T 0,28,0,945,0,1260,0,378,0,36,945,0,4725,0,3150,0,630,0,45
%N Triangle read by rows: T(n,k) (n>=2, 0 <= k <= n-2) = n!/(2^i*i!*k!), where k=n-2i (or 0 for entries with wrong parity).
%H J. East and R. D. Gray, <a href="http://arxiv.org/abs/1404.2359">Idempotent generators in finite partition monoids and related semigroups</a>, arXiv preprint arXiv:1404.2359 [math.GR], 2014-2016.
%e Triangle begins:
%e 1;
%e 0, 3;
%e 3, 0, 6;
%e 0, 15, 0, 10;
%e 15, 0, 45, 0, 15;
%e 0, 105, 0, 105, 0, 21;
%e 105, 0, 420, 0, 210, 0, 28;
%e 0, 945, 0, 1260, 0, 378, 0, 36;
%e 945, 0, 4725, 0, 3150, 0, 630, 0, 45;
%e ...
%t T[n_, k_] := With[{i = (n-k)/2}, If[EvenQ[n-k], n!/(2^i i! k!), 0]];
%t Table[T[n, k], {n, 2, 10}, {k, 0, n-2}] // Flatten (* _Jean-François Alcover_, Nov 25 2018 *)
%Y This is A099174 without the two rightmost diagonals.
%K nonn,tabl
%O 0,3
%A _N. J. A. Sloane_, Jul 05 2014
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