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A244492 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).

%I A244492 #23 Jul 16 2022 11:49:11
%S A244492 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 A244492 0,28,0,945,0,1260,0,378,0,36,945,0,4725,0,3150,0,630,0,45
%N A244492 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 A244492 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 A244492 Triangle begins:
%e A244492     1;
%e A244492     0,   3;
%e A244492     3,   0,    6;
%e A244492     0,  15,    0,   10;
%e A244492    15,   0,   45,    0,   15;
%e A244492     0, 105,    0,  105,    0,  21;
%e A244492   105,   0,  420,    0,  210,   0,  28;
%e A244492     0, 945,    0, 1260,    0, 378,   0, 36;
%e A244492   945,   0, 4725,    0, 3150,   0, 630,  0, 45;
%e A244492   ...
%t A244492 T[n_, k_] := With[{i = (n-k)/2}, If[EvenQ[n-k], n!/(2^i i! k!), 0]];
%t A244492 Table[T[n, k], {n, 2, 10}, {k, 0, n-2}] // Flatten (* _Jean-François Alcover_, Nov 25 2018 *)
%Y A244492 This is A099174 without the two rightmost diagonals.
%K A244492 nonn,tabl
%O A244492 0,3
%A A244492 _N. J. A. Sloane_, Jul 05 2014