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%I #62 Apr 06 2020 18:34:26
%S 1,1,1,1,3,1,1,6,6,1,1,10,21,10,1,1,15,56,55,15,1,1,21,126,220,120,21,
%T 1,1,28,252,715,680,231,28,1,1,36,462,2002,3060,1771,406,36,1,1,45,
%U 792,5005,11628,10626,4060,666,45,1,1,55,1287,11440,38760,53130,31465,8436
%N Triangle of triangular binomial coefficients, read by rows, where column k has the g.f.: 1/(1-x)^((k+1)*(k+2)/2) for k >= 0.
%C The row sums form A098569: {1,2,5,14,43,143,510,1936,7775,32869,...}. How do the terms of row k tend to be distributed as k grows?
%C Remarkably, column k of the matrix inverse (A121434) equals signed column k of the triangular matrix power: A107876^(k*(k+1)/2) for k >= 0. - _Paul D. Hanna_, Aug 25 2006
%C Surprisingly, the row sums (A098569) equal the row sums of triangle A131338. - _Paul D. Hanna_, Aug 30 2007
%C Number of sequences S = s(1)s(2)...s(n) such that S contains m 0's, for 1<=j<=n, s(j)<j and s(j-s(j)) = 0, for 1 < j <= n, if s(j) positive, then s(j-1) < s(j). - _Frank Ruskey_, Apr 15 2011
%C As a rectangular array read by antidiagonals R(n,k) (n>=2, k>=0) is the number of labeled graphs on n nodes that have exactly k arcs where multiple arcs are allowed to connect distinct vertex pairs. R(n,k) = C(C(n,2)+k-1,k). See example below. - _Geoffrey Critzer_, Nov 12 2011
%H Paul D. Hanna, <a href="/A098568/b098568.txt">Table of n, a(n) for n = 0..1080, of flattened triangle, read by rows 0..45.</a>
%H Soheir M. Khamis, <a href="http://dx.doi.org/10.1016/S0012-365X(03)00106-7">Height counting of unlabeled interval and N-free posets</a>, Discrete Math. 275 (2004), no. 1-3, 165-175.
%H Nate Kube and Frank Ruskey, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL8/Ruskey/ruskey99.html">Sequences That Satisfy a(n-a(n))=0</a>, Journal of Integer Sequences, Vol. 8 (2005), Article 05.5.5.
%H Zhicong Lin and Shishuo Fu, <a href="https://arxiv.org/abs/2003.11813">On 120-avoiding inversion and ascent sequences</a>, arXiv:2003.11813 [math.CO], 2020.
%H R. J. Mathar, <a href="http://arxiv.org/abs/1709.09000">Statistics on Small Graphs</a>, arXiv:1709.09000 [math.CO], (2017), table 60.
%F T(n, k) = binomial((k+1)*(k+2)/2 + n-k-1, n-k).
%e G.f.s of columns: 1/(1-x), 1/(1-x)^3, 1/(1-x)^6, 1/(1-x)^10, 1/(1-x)^15, ...
%e Rows begin:
%e 1;
%e 1, 1;
%e 1, 3, 1;
%e 1, 6, 6, 1;
%e 1, 10, 21, 10, 1;
%e 1, 15, 56, 55, 15, 1;
%e 1, 21, 126, 220, 120, 21, 1;
%e 1, 28, 252, 715, 680, 231, 28, 1;
%e 1, 36, 462, 2002, 3060, 1771, 406, 36, 1;
%e 1, 45, 792, 5005, 11628, 10626, 4060, 666, 45, 1;
%e 1, 55, 1287, 11440, 38760, 53130, 31465, 8436, 1035, 55, 1;
%e 1, 66, 2002, 24310, 116280, 230230, 201376, 82251, 16215, 1540, 66, 1; ...
%e From _Frank Ruskey_, Apr 15 2011: (Start)
%e In reference to comment about s(1)s(2)...s(n) above,
%e a(4,2) = 6 = |{0012, 0013, 0023, 0101, 0103, 0120}| and
%e a(4,3) = 6 = |{0001, 0002, 0003, 0010, 0020, 0100}|. (End)
%e From _Geoffrey Critzer_, Nov 12 2011: (Start)
%e In reference to comment about multigraphs above,
%e 1, 1, 1, 1, 1, 1, ... 2 nodes
%e 1, 3, 6, 10, 15, 21, ... 3 nodes
%e 1, 6, 21, 56, 126, 252, ... .
%e 1, 10, 55, 220, 715, 2002, ... .
%e 1, 15, 120, 680, 3060, 11628, ... .
%e 1, 21, 231, 1771, 10626, 58130, ... . (End)
%t t[n_, k_] = Binomial[(k+1)*(k+2)/2 + n-k-1, n-k]; Flatten[Table[t[n, k], {n, 0, 10}, {k, 0, n}]] (* _Jean-François Alcover_, Jul 18 2011 *)
%o (PARI) {T(n,k)=binomial((k+1)*(k+2)/2+n-k-1,n-k)}
%o for(n=0,12,for(k=0,n,print1(T(n,k),", "));print(""))
%Y Cf. A098569. A290428 (unlabeled graphs).
%Y Cf. A121434 (inverse); variants: A122175, A122176, A122177; A107876.
%Y Cf. A131338.
%K nonn,tabl
%O 0,5
%A _Paul D. Hanna_, Sep 15 2004
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