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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.
13
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, 1, 1, 28, 252, 715, 680, 231, 28, 1, 1, 36, 462, 2002, 3060, 1771, 406, 36, 1, 1, 45, 792, 5005, 11628, 10626, 4060, 666, 45, 1, 1, 55, 1287, 11440, 38760, 53130, 31465, 8436
OFFSET
0,5
COMMENTS
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?
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
Surprisingly, the row sums (A098569) equal the row sums of triangle A131338. - Paul D. Hanna, Aug 30 2007
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
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
LINKS
Soheir M. Khamis, Height counting of unlabeled interval and N-free posets, Discrete Math. 275 (2004), no. 1-3, 165-175.
Nate Kube and Frank Ruskey, Sequences That Satisfy a(n-a(n))=0, Journal of Integer Sequences, Vol. 8 (2005), Article 05.5.5.
Zhicong Lin and Shishuo Fu, On 120-avoiding inversion and ascent sequences, arXiv:2003.11813 [math.CO], 2020.
R. J. Mathar, Statistics on Small Graphs, arXiv:1709.09000 [math.CO], (2017), table 60.
FORMULA
T(n, k) = binomial((k+1)*(k+2)/2 + n-k-1, n-k).
EXAMPLE
G.f.s of columns: 1/(1-x), 1/(1-x)^3, 1/(1-x)^6, 1/(1-x)^10, 1/(1-x)^15, ...
Rows begin:
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, 1;
1, 28, 252, 715, 680, 231, 28, 1;
1, 36, 462, 2002, 3060, 1771, 406, 36, 1;
1, 45, 792, 5005, 11628, 10626, 4060, 666, 45, 1;
1, 55, 1287, 11440, 38760, 53130, 31465, 8436, 1035, 55, 1;
1, 66, 2002, 24310, 116280, 230230, 201376, 82251, 16215, 1540, 66, 1; ...
From Frank Ruskey, Apr 15 2011: (Start)
In reference to comment about s(1)s(2)...s(n) above,
a(4,2) = 6 = |{0012, 0013, 0023, 0101, 0103, 0120}| and
a(4,3) = 6 = |{0001, 0002, 0003, 0010, 0020, 0100}|. (End)
From Geoffrey Critzer, Nov 12 2011: (Start)
In reference to comment about multigraphs above,
1, 1, 1, 1, 1, 1, ... 2 nodes
1, 3, 6, 10, 15, 21, ... 3 nodes
1, 6, 21, 56, 126, 252, ... .
1, 10, 55, 220, 715, 2002, ... .
1, 15, 120, 680, 3060, 11628, ... .
1, 21, 231, 1771, 10626, 58130, ... . (End)
MATHEMATICA
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 *)
PROG
(PARI) {T(n, k)=binomial((k+1)*(k+2)/2+n-k-1, n-k)}
for(n=0, 12, for(k=0, n, print1(T(n, k), ", ")); print(""))
CROSSREFS
Cf. A098569. A290428 (unlabeled graphs).
Cf. A121434 (inverse); variants: A122175, A122176, A122177; A107876.
Cf. A131338.
Sequence in context: A107105 A088925 A100862 * A180959 A131235 A202812
KEYWORD
nonn,tabl
AUTHOR
Paul D. Hanna, Sep 15 2004
STATUS
approved