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A241188 Triangle T(n,s) of Dynkin type D_n read by rows (n >= 2, 0 <= s <= n).

%I A241188 #30 Jun 03 2019 09:07:25
%S A241188 1,2,1,1,3,5,5,1,4,9,16,20,1,5,14,30,55,77,1,6,20,50,105,196,294,1,7,
%T A241188 27,77,182,378,714,1122,1,8,35,112,294,672,1386,2640,4290,1,9,44,156,
%U A241188 450,1122,2508,5148,9867,16445
%N A241188 Triangle T(n,s) of Dynkin type D_n read by rows (n >= 2, 0 <= s <= n).
%H A241188 M. A. A. Obaid, S. K. Nauman, W. M. Fakieh, C. M. Ringel, <a href="http://www.math.uni-bielefeld.de/~ringel/opus/jeddah.pdf">The numbers of support-tilting modules for a Dynkin algebra</a>, 2014.
%H A241188 M. A. A. Obaid, S. K. Nauman, W. M. Fakieh, C. M. Ringel, <a href="http://arxiv.org/abs/1403.5827">The numbers of support-tilting modules for a Dynkin algebra</a>, arXiv:1403.5827 [math.RT], 2014  and <a href="https://cs.uwaterloo.ca/journals/JIS/VOL18/Ringel/ringel22.html">J. Int. Seq. 18 (2015) 15.10.6</a>.
%F A241188 T(n,s) = [n+s-2,s] for 0 <= s < n, T(n,n) = [2n-2,n-2], where [t,s] stands for binomial(t,s)*(s+t)/t.
%e A241188 Triangle begins:
%e A241188 1, 2, 1,
%e A241188 1, 3, 5, 5,
%e A241188 1, 4, 9, 16, 20,
%e A241188 1, 5, 14, 30, 55, 77,
%e A241188 1, 6, 20, 50, 105, 196, 294,
%e A241188 1, 7, 27, 77, 182, 378, 714, 1122,
%e A241188 1, 8, 35, 112, 294, 672, 1386, 2640, 4290,
%e A241188 1, 9, 44, 156, 450, 1122, 2508, 5148, 9867, 16445,
%e A241188 ...
%t A241188 f[t_, s_] := Binomial[t, s] (s + t)/t;
%t A241188 T[_, 0] = 1; T[n_, n_] := f[2 n - 2, n - 2]; T[n_, s_] := f[n + s - 2, s];
%t A241188 Table[T[n, s], {n, 2, 9}, {s, 0, n}] // Flatten (* _Jean-François Alcover_, Feb 12 2019 *)
%Y A241188 See A009766 for the case of type A.
%Y A241188 See A059481 for the case of type B/C.
%Y A241188 Diagonals give A029869, A051960, A029651, A051924. Row sums are also A051924.
%K A241188 nonn,tabf
%O A241188 2,2
%A A241188 _N. J. A. Sloane_, Apr 24 2014