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A098568 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.

%I A098568 #62 Apr 06 2020 18:34:26
%S A098568 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 A098568 1,1,28,252,715,680,231,28,1,1,36,462,2002,3060,1771,406,36,1,1,45,
%U A098568 792,5005,11628,10626,4060,666,45,1,1,55,1287,11440,38760,53130,31465,8436
%N A098568 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 A098568 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 A098568 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 A098568 Surprisingly, the row sums (A098569) equal the row sums of triangle A131338. - _Paul D. Hanna_, Aug 30 2007
%C A098568 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 A098568 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 A098568 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 A098568 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 A098568 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 A098568 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 A098568 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 A098568 T(n, k) = binomial((k+1)*(k+2)/2 + n-k-1, n-k).
%e A098568 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 A098568 Rows begin:
%e A098568   1;
%e A098568   1,  1;
%e A098568   1,  3,    1;
%e A098568   1,  6,    6,     1;
%e A098568   1, 10,   21,    10,      1;
%e A098568   1, 15,   56,    55,     15,      1;
%e A098568   1, 21,  126,   220,    120,     21,      1;
%e A098568   1, 28,  252,   715,    680,    231,     28,     1;
%e A098568   1, 36,  462,  2002,   3060,   1771,    406,    36,     1;
%e A098568   1, 45,  792,  5005,  11628,  10626,   4060,   666,    45,    1;
%e A098568   1, 55, 1287, 11440,  38760,  53130,  31465,  8436,  1035,   55,  1;
%e A098568   1, 66, 2002, 24310, 116280, 230230, 201376, 82251, 16215, 1540, 66, 1; ...
%e A098568 From _Frank Ruskey_, Apr 15 2011: (Start)
%e A098568 In reference to comment about s(1)s(2)...s(n) above,
%e A098568    a(4,2) = 6 = |{0012, 0013, 0023, 0101, 0103, 0120}|  and
%e A098568    a(4,3) = 6 = |{0001, 0002, 0003, 0010, 0020, 0100}|. (End)
%e A098568 From _Geoffrey Critzer_, Nov 12 2011: (Start)
%e A098568 In reference to comment about multigraphs above,
%e A098568   1,    1,    1,    1,    1,     1,     ...  2 nodes
%e A098568   1,    3,    6,    10,   15,    21,    ...  3 nodes
%e A098568   1,    6,    21,   56,   126,   252,   ...  .
%e A098568   1,    10,   55,   220,  715,   2002,  ...  .
%e A098568   1,    15,   120,  680,  3060,  11628, ...  .
%e A098568   1,    21,   231,  1771, 10626, 58130, ...  . (End)
%t A098568 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 A098568 (PARI) {T(n,k)=binomial((k+1)*(k+2)/2+n-k-1,n-k)}
%o A098568 for(n=0,12,for(k=0,n,print1(T(n,k),", "));print(""))
%Y A098568 Cf. A098569. A290428 (unlabeled graphs).
%Y A098568 Cf. A121434 (inverse); variants: A122175, A122176, A122177; A107876.
%Y A098568 Cf. A131338.
%K A098568 nonn,tabl
%O A098568 0,5
%A A098568 _Paul D. Hanna_, Sep 15 2004