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.

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

Paul D. Hanna, Table of n, a(n) for n = 0..1080, of flattened triangle, read by rows 0..45.

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

Adjacent sequences: A098565 A098566 A098567 * A098569 A098570 A098571

Keyword

nonn,tabl

Author

Paul D. Hanna, Sep 15 2004

Status

approved