A098568 -id:A098568 - cp's OEIS frontend

A098569 Row sums of the triangle of triangular binomial coefficients given by A098568.

1, 2, 5, 14, 43, 143, 510, 1936, 7775, 32869, 145665, 674338, 3251208, 16282580, 84512702, 453697993, 2514668492, 14367066833, 84489482201, 510760424832, 3170267071640, 20182121448815, 131642848217536, 878999194493046, 6003048930287115, 41899203336942661

Offset: 0

Paul D. Hanna, Sep 15 2004, Jun 29 2007

From Lara Pudwell, Oct 23 2008: (Start)
A permutation p avoids a pattern q if it has no subsequence that is order-isomorphic to q. For example, p avoids the pattern 132 if it has no subsequence abc with a < c < b.
Barred pattern avoidance considers permutations that avoid a pattern except in a special case. Given a barred pattern q, we may form two patterns, q1 = the sequence of unbarred letters of q and q2 = the sequence of all letters of q.
A permutation p avoids barred pattern q if every instance of q1 in p is embedded in a copy of q2 in p. In other words, p avoids q1, except in the special case that a copy of q1 is a subsequence of a copy of q2.
For example, if q=5{bar 1}32{bar 4}, then q1=532 and q2 = 51324. p avoids q if every for decreasing subsequence acd of length 3 in p, one can find letters b and e so that the subsequence abcde of p has b < d < c < e < a.
(End)
Also equals the row sums of triangle A131338, which starts with a '1' in row 0 and then for n > 0 row n consists of n '1's followed by the partial sums of the prior row.
Also the number of permutations in S_n avoiding {bar 4}25{bar 1}3 (i.e., every occurrence of 253 is contained in an occurrence of a 42513). - Lara Pudwell, Apr 25 2008 (see the Claesson-Dukes-Kitaev article)
From Frank Ruskey, Apr 17 2011: (Start)
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).
(End)
a(n) is also the number of length n permutations that simultaneously avoid the bivincular patterns (132,{2},{}) and (132,{},{2}). - Christian Bean, Mar 25 2015
a(n) is also the number of length n permutations that simultaneously avoid the bivincular patterns (123,{2},{}) and (123,{},{2}). These are the same as the permutations avoiding {bar 4}23{bar 1}5. - Christian Bean, Jun 03 2015
From Peter R. W. McNamara, Jun 22 2019: (Start)
a(n) is the number of upper-triangular matrices with nonnegative integer entries whose entries sum to n, and whose diagonal entries are all positive.
a(n) is the number of ascent sequences [d(1), d(2), ..., d(n)] A022493 for which d(k) comes from the interval [0, d(k-1)] or equals 1 + max([d(1), d(2), ..., d(k-1)]) = 1 + asc([d(1), d(2), ..., d(k-1)]) where asc(.) counts the ascents of its argument.  Such sequences are called "self modified ascent sequences" in Bousquet-Mélou et al.
The elements of a (2+2)-free poset can be partitioned into levels, where all elements at the same level have the same strict down-set.  Then a(n) is the number of unlabeled (2+2)-free posets with n elements that contain a chain with exactly one element at each level.
(End)

			In reference to comment about s(1)s(2)...s(n) above, a(3) = 14 = |{0000, 0001, 0002, 0003, 0010, 0020, 0100, 0012, 0013, 0023, 0101, 0103, 0120, 0123}|. - _Frank Ruskey_, Apr 17 2011
From _Paul D. Hanna_, Aug 24 2025: (Start)
The following array (A131338) illustrates a process that generates these numbers. Start with [1] in row n = 0. For n > 0, form row n by concatenating n 1's with the partial sums of the prior row. The row sums of row n equals a(n) for n >= 0; equivalently, the final term of row n+1 equals a(n). Continuing in this way generates all the terms of this sequence.
  n = 0; [1];
  n = 1: [1, 1];
  n = 2; [1, 1, 1, 2];
  n = 3: [1, 1, 1, 1, 2, 3, 5];
  n = 4: [1, 1, 1, 1, 1, 2, 3, 4, 6, 9, 14];
  n = 5: [1, 1, 1, 1, 1, 1, 2, 3, 4, 5, 7, 10, 14, 20, 29, 43];
  n = 6: [1, 1, 1, 1, 1, 1, 1, 2, 3, 4, 5, 6, 8, 11, 15, 20, 27, 37, 51, 71, 100, 143];
  n = 7: [1, 1, 1, 1, 1, 1, 1, 1, 2, 3, 4, 5, 6, 7, 9, 12, 16, 21, 27, 35, 46, 61, 81, 108, 145, 196, 267, 367, 510];
  ... (End)

Andrew Howroyd, Table of n, a(n) for n = 0..500
Christian Bean, A. Claesson and H. Ulfarsson, Simultaneous Avoidance of a Vincular and a Covincular Pattern of Length 3, arXiv preprint arXiv:1512.03226 [math.CO], 2015-2017.
Beáta Bényi, Toufik Mansour, and José L. Ramírez, Pattern Avoidance in Weak Ascent Sequences, arXiv:2309.06518 [math.CO], 2023.
Mireille Bousquet-Mélou, Anders Claesson, Mark Dukes and Sergey Kitaev, (2+2)-free posets, ascent sequences and pattern avoiding permutations, arXiv:0806.0666 [math.CO], 2008-2009.
William Y. C. Chen, Alvin Y.L. Dai, Theodore Dokos, Tim Dwyer and Bruce E. Sagan, On 021-Avoiding Ascent Sequences, The Electronic Journal of Combinatorics Volume 20, Issue 1 (2013), #P76.
CombOS - Combinatorial Object Server, Generate pattern-avoiding permutations
Mark Dukes and Peter R. W. McNamara, Refining the bijections among ascent sequences, (2+2)-free posets, integer matrices and pattern-avoiding permutations, arXiv:1807.11505 [math.CO], 2018-2019; Journal of Combinatorial Theory (Series A), 167 (2019), 403-430.
Elizabeth Hartung, Hung Phuc Hoang, Torsten Mütze and Aaron Williams, Combinatorial generation via permutation languages. I. Fundamentals, arXiv:1906.06069 [cs.DM], 2019.
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 Sherry H. F. Yan, Vincular patterns in inversion sequences, Applied Mathematics and Computation (2020), Vol. 364, 124672.
Lara Pudwell, Enumeration Schemes for Pattern-Avoiding Words and Permutations, Ph. D. Dissertation, Math. Dept., Rutgers University, May 2008.
Lara Pudwell, Enumeration schemes for permutations avoiding barred patterns, El. J. Combinat. 17 (1) (2010) R29.

Cf. A098568, A131338.

A098569 := proc(n)
    add( binomial((k+1)*(k+2)/2+n-k-1,n-k),k=0..n) ;
end proc:
seq(A098569(n),n=0..40) ; # Georg Fischer, Oct 29 2023

Table[Sum[Binomial[(k+1)*(k+2)/2+n-k-1, n-k],{k,0,n}],{n,0,20}] (* Vaclav Kotesovec, Apr 05 2015 *)

a(n)=sum(k=0,n,binomial((k+1)*(k+2)/2+n-k-1,n-k))

a(n) = Sum_{k=0..n} C( (k+1)*(k+2)/2 + n-k-1, n-k).
G.f: Sum_{k>=0} x^k*y^C(k+1,2) where y = 1/(1-x). - Christian Bean, Mar 25 2015
log(a(n)) ~ n*(log(n) - 2*log(log(n)) + log(2) - 1 + 4*log(log(n))/log(n) - 2*log(2)/log(n) - 2/log(n)^2). - Vaclav Kotesovec, Oct 30 2023

Offset changed to 0 by Georg Fischer, Oct 29 2023

A121434 Matrix inverse of triangle A098568, where A098568(n, k) = C( (k+1)*(k+2)/2 + n-k-1, n-k) for n>=k>=0.

Original entry on oeis.org

1, 0, 1, 0, -1, 1, 0, 2, -3, 1, 0, -7, 12, -6, 1, 0, 37, -67, 39, -10, 1, 0, -268, 498, -311, 95, -15, 1, 0, 2496, -4701, 3045, -1015, 195, -21, 1, 0, -28612, 54298, -35901, 12560, -2675, 357, -28, 1, 0, 391189, -745734, 499157, -179717, 40635, -6097, 602, -36, 1, 0, -6230646, 11911221, -8034267, 2945010

Offset: 0

Paul D. Hanna, Aug 27 2006

			Triangle begins:
  1;
  0, 1;
  0, -1, 1;
  0, 2, -3, 1;
  0, -7, 12, -6, 1;
  0, 37, -67, 39, -10, 1;
  0, -268, 498, -311, 95, -15, 1;
  0, 2496, -4701, 3045, -1015, 195, -21, 1;
  0, -28612, 54298, -35901, 12560, -2675, 357, -28, 1;
  0, 391189, -745734, 499157, -179717, 40635, -6097, 602, -36, 1; ...

Cf. A098568, A107876; unsigned columns: A107877, A107887.

/* Matrix Inverse of A098568 */ T(n,k)=local(M=matrix(n+1,n+1,r,c,if(r>=c,binomial((c-1)*(c-2)/2+r-2,r-c)))); return((M^-1)[n+1,k+1])

/* Obtain by G.F. */ T(n,k)=polcoeff(1-sum(j=0, n-k-1, T(j+k,k)*x^j/(1-x+x*O(x^n))^(j*(j+1)/2+j*k+k*(k+1)/2)), n-k)

(1) T(n,k) = (-1)^(n-k)*[A107876^(k*(k+1)/2)](n,k); i.e., column k equals signed column k of A107876^(k*(k+1)/2).
G.f.s for column k:
(2) 1 = Sum_{j>=0} T(j+k,k)*x^j/(1-x)^( j*(j+1)/2) + j*k + k*(k+1)/2);
(3) 1 = Sum_{j>=0} T(j+k,k)*x^j*(1+x)^( j*(j-1)/2) + j*k + k*(k+1)/2).

A131338 Triangle, read by rows of n*(n+1)/2 + 1 terms, that starts with a '1' in row 0 with row n consisting of n '1's followed by the partial sums of the prior row.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 2, 3, 5, 1, 1, 1, 1, 1, 2, 3, 4, 6, 9, 14, 1, 1, 1, 1, 1, 1, 2, 3, 4, 5, 7, 10, 14, 20, 29, 43, 1, 1, 1, 1, 1, 1, 1, 2, 3, 4, 5, 6, 8, 11, 15, 20, 27, 37, 51, 71, 100, 143, 1, 1, 1, 1, 1, 1, 1, 1, 2, 3, 4, 5, 6, 7, 9, 12, 16, 21, 27, 35, 46, 61, 81, 108, 145, 196

Offset: 0

Paul D. Hanna, Jun 29 2007

			Triangle begins:
1;
1, 1;
1,1, 1,2;
1,1,1, 1,2,3,5;
1,1,1,1, 1,2,3,4,6,9,14;
1,1,1,1,1, 1,2,3,4,5,7,10,14,20,29,43;
1,1,1,1,1,1, 1,2,3,4,5,6,8,11,15,20,27,37,51,71,100,143;
1,1,1,1,1,1,1, 1,2,3,4,5,6,7,9,12,16,21,27,35,46,61,81,108,145,196,267,367,510; ...
Row sums equal the row sums (A098569) of triangle A098568,
where A098568(n, k) = binomial( (k+1)*(k+2)/2 + n-k-1, n-k):
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; ...

Paul D. Hanna, Rows n = 0..16, flattened.

Cf. A098568, A098569 (row sums), A121690, A183202.

Cf. A214403 (variant).

T(n,k)=if(k>n*(n+1)/2 || k<0,0,if(k<=n,1,sum(i=0,k-n,T(n-1,i))))
for(n=0, 10, for(k=0, n*(n+1)/2, print1(T(n, k), ", ")); print(""))

T(n,k) = Sum_{i=0..k-n} T(n-1,i) for k>n, else T(n,k)=1 for n>=k>=0.
Right border: T(n+1, (n+1)*(n+2)/2) = A098569(n) = Sum_{k=0..n} C( (k+1)*(k+2)/2 + n-k-1, n-k).
T(n, n*(n-1)/2 + 1) = Sum_{k=0..n-1} C(k*(k+1)/2, n-k) = A121690(n-1) for n>=1. - Paul D. Hanna, Aug 30 2007

A290428 Array read by antidiagonals: T(n,k) is the number of graphs with n edges and k vertices, allowing loops and multi-edges.

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 1, 2, 1, 0, 1, 2, 4, 1, 0, 1, 2, 6, 6, 1, 0, 1, 2, 7, 14, 9, 1, 0, 1, 2, 7, 20, 28, 12, 1, 0, 1, 2, 7, 22, 53, 52, 16, 1, 0, 1, 2, 7, 23, 69, 125, 93, 20, 1, 0, 1, 2, 7, 23, 76, 198, 287, 152, 25, 1, 0, 1, 2, 7, 23, 78, 245, 550, 606, 242, 30, 1, 0

Offset: 0

R. J. Mathar, Jul 31 2017

Variant of A138107, here for non-directed edges.

			1   1   1   1    1    1    1   1   1...
0   1   2   2    2    2    2   2   2...
0   1   4   6    7    7    7   7   7...
0   1   6  14   20   22   23  23  23...
0   1   9  28   53   69   76  78  79...
0   1  12  52  125  198  245 264 271...
0   1  16  93  287  550  782 915 973...
0   1  20 152  606 1441 2392
0   1  25 242 1226 3611

Andrew Howroyd, Table of n, a(n) for n = 0..1325
R. J. Mathar, Statistics on Small Graphs, arXiv:1709.09000 [math.CO], 2017. See Table 59.

Cf. A050531 (column 3), A050532 (column 4), A138107, A098568 (vertex-labeled).

rows = 12;
permcount[v_] := Module[{m=1, s=0, k=0, t}, For[i=1, i <= Length[v], i++, t = v[[i]]; k = If[i>1 && t == v[[i-1]], k+1, 1]; m *= t*k; s += t]; s!/m];
edges[v_, t_] := Product[g = GCD[v[[i]], v[[j]] ]; t[v[[i]]*v[[j]]/g]^g, {i, 2, Length[v]}, {j, 1, i - 1}]*Product[c = v[[i]]; t[c]^Quotient[c + 1, 2]*If[OddQ[c], 1, t[c/2]], {i, 1, Length[v]}];
col[k_] := col[k] = Module[{s = O[x]^rows}, Do[s += permcount[p]*1/edges[p, 1 - x^# + O[x]^rows&], {p, IntegerPartitions[k]}]; s/k!] // CoefficientList[#, x]&;
T[0, ] = 1; T[, 0] = 0;
T[n_, k_] := col[k][[n + 1]];
Table[T[n-k, k], {n, 0, rows-1}, {k, n, 0, -1}] // Flatten (* Jean-François Alcover, Jan 08 2021, after Andrew Howroyd *)

permcount(v) = {my(m=1,s=0,k=0,t); for(i=1,#v,t=v[i]; k=if(i>1&&t==v[i-1],k+1,1); m*=t*k;s+=t); s!/m}
edges(v,t) = {prod(i=2, #v, prod(j=1, i-1, my(g=gcd(v[i],v[j])); t(v[i]*v[j]/g)^g )) * prod(i=1, #v, my(c=v[i]); t(c)^((c+1)\2)*if(c%2, 1, t(c/2)))}
T(m, n=m) = {Mat(vector(n+1, n, my(s=O(x*x^m)); forpart(p=n-1, s+=permcount(p)*1/edges(p,i->1-x^i+O(x*x^m))); Col(s/(n-1)!)))}
{ my(A=T(8)); for(n=1, #A, print(A[n,])) } \\ Andrew Howroyd, Oct 22 2019

A122176 Triangle, read by rows, where T(n,k) = C( k*(k+1)/2 + n-k + 1, n-k) for n>=k>=0.

Original entry on oeis.org

1, 2, 1, 3, 3, 1, 4, 6, 5, 1, 5, 10, 15, 8, 1, 6, 15, 35, 36, 12, 1, 7, 21, 70, 120, 78, 17, 1, 8, 28, 126, 330, 364, 153, 23, 1, 9, 36, 210, 792, 1365, 969, 276, 30, 1, 10, 45, 330, 1716, 4368, 4845, 2300, 465, 38, 1, 11, 55, 495, 3432, 12376, 20349, 14950, 4960, 741, 47, 1

Offset: 0

Paul D. Hanna, Aug 25 2006

Remarkably, column k of the matrix inverse (A121436) equals signed column k of matrix power: A107876^(k*(k+1)/2 + 2).

			Triangle begins:
1;
2, 1;
3, 3, 1;
4, 6, 5, 1;
5, 10, 15, 8, 1;
6, 15, 35, 36, 12, 1;
7, 21, 70, 120, 78, 17, 1;
8, 28, 126, 330, 364, 153, 23, 1;
9, 36, 210, 792, 1365, 969, 276, 30, 1; ...

Harvey P. Dale, Table of n, a(n) for n = 0..10000

Cf. A121436 (inverse); variants: A098568, A122175, A122177.

Flatten[Table[Binomial[(k(k+1))/2+n-k+1,n-k],{n,0,10},{k,0,n}]] (* Harvey P. Dale, Mar 18 2013 *)

PARI
```
T(n,k)=if(n
				
```

A122177 Triangle, read by rows, where T(n,k) = C( k*(k+1)/2 + n-k + 2, n-k) for n>=k>=0.

Original entry on oeis.org

1, 3, 1, 6, 4, 1, 10, 10, 6, 1, 15, 20, 21, 9, 1, 21, 35, 56, 45, 13, 1, 28, 56, 126, 165, 91, 18, 1, 36, 84, 252, 495, 455, 171, 24, 1, 45, 120, 462, 1287, 1820, 1140, 300, 31, 1, 55, 165, 792, 3003, 6188, 5985, 2600, 496, 39, 1, 66, 220, 1287, 6435, 18564, 26334

Offset: 0

Paul D. Hanna, Aug 25 2006

Remarkably, column k of the matrix inverse (A121437) equals signed column k of matrix power: A107876^(k*(k+1)/2 + 3) for k>=0.

			Triangle begins:
1;
3, 1;
6, 4, 1;
10, 10, 6, 1;
15, 20, 21, 9, 1;
21, 35, 56, 45, 13, 1;
28, 56, 126, 165, 91, 18, 1;
36, 84, 252, 495, 455, 171, 24, 1;
45, 120, 462, 1287, 1820, 1140, 300, 31, 1; ...

Cf. A121437 (inverse); variants: A098568, A122175, A122176.

A122177[n_, k_] := Binomial[k*(k + 1)/2 + n - k + 2, n - k];
Table[A122177[n, k], {n, 0, 10}, {k, 0, n}] (* Paolo Xausa, Jul 23 2024 *)

PARI
```
T(n,k)=if(n
				
```

A122175 Triangle, read by rows, where T(n,k) = C( k*(k+1)/2 + n-k, n-k) for n>=k>=0.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 3, 4, 1, 1, 4, 10, 7, 1, 1, 5, 20, 28, 11, 1, 1, 6, 35, 84, 66, 16, 1, 1, 7, 56, 210, 286, 136, 22, 1, 1, 8, 84, 462, 1001, 816, 253, 29, 1, 1, 9, 120, 924, 3003, 3876, 2024, 435, 37, 1, 1, 10, 165, 1716, 8008, 15504, 12650, 4495, 703, 46, 1, 1, 11, 220

Offset: 0

Paul D. Hanna, Aug 25 2006

Remarkably, column k of the matrix inverse (A121435) equals signed column k of matrix power: A107876^(k*(k+1)/2 + 1).

			Triangle begins:
1;
1, 1;
1, 2, 1;
1, 3, 4, 1;
1, 4, 10, 7, 1;
1, 5, 20, 28, 11, 1;
1, 6, 35, 84, 66, 16, 1;
1, 7, 56, 210, 286, 136, 22, 1;
1, 8, 84, 462, 1001, 816, 253, 29, 1; ...

Cf. A121435 (inverse); variants: A098568, A122176, A122177.

Table[Binomial[(k(k+1))/2+n-k,n-k],{n,0,20},{k,0,n}]//Flatten (* Harvey P. Dale, Mar 22 2016 *)

PARI
```
T(n,k)=if(n
				
```

A122178 Triangle, read by rows, where T(n,k) = C( n*(n+1)/2 + n-k - 1, n-k), for n>=k>=0.

Original entry on oeis.org

1, 1, 1, 6, 3, 1, 56, 21, 6, 1, 715, 220, 55, 10, 1, 11628, 3060, 680, 120, 15, 1, 230230, 53130, 10626, 1771, 231, 21, 1, 5379616, 1107568, 201376, 31465, 4060, 406, 28, 1, 145008513, 26978328, 4496388, 658008, 82251, 8436, 666, 36, 1, 4431613550

Offset: 0

Paul D. Hanna, Aug 29 2006

A triangle having similar properties and complementary construction is the dual triangle A098568.

			Triangle begins:
1;
1, 1;
6, 3, 1;
56, 21, 6, 1;
715, 220, 55, 10, 1;
11628, 3060, 680, 120, 15, 1;
230230, 53130, 10626, 1771, 231, 21, 1;
5379616, 1107568, 201376, 31465, 4060, 406, 28, 1;
145008513, 26978328, 4496388, 658008, 82251, 8436, 666, 36, 1; ...

Cf. A121438 (matrix inverse); A121412; variants: A121334, A121335, A121336; A098568 (dual).

PARI
```
T(n,k)=binomial(n*(n+1)/2+n-k-1,n-k)
```

Remarkably, row n of the matrix inverse (A121438) equals row n of A121412^(-n*(n+1)/2). Further, the following matrix products of triangles of binomial coefficients are equal: A121412 = A121334*A122178^-1 = A121335*A121334^-1 = A121336*A121335^-1, where row n of H=A121412 equals row (n-1) of H^(n+1) with an appended '1'.

A121436 Matrix inverse of triangle A122176, where A122176(n,k) = C( k*(k+1)/2 + n-k + 1, n-k) for n>=k>=0.

Original entry on oeis.org

1, -2, 1, 3, -3, 1, -7, 9, -5, 1, 26, -37, 25, -8, 1, -141, 210, -155, 60, -12, 1, 1034, -1575, 1215, -516, 126, -17, 1, -9693, 14943, -11806, 5270, -1426, 238, -23, 1, 111522, -173109, 138660, -63696, 18267, -3417, 414, -30, 1, -1528112, 2381814, -1923765, 899226, -267084, 53431, -7337, 675, -38, 1

Offset: 0

Paul D. Hanna, Aug 27 2006

			Triangle begins:
1;
-2, 1;
3, -3, 1;
-7, 9, -5, 1;
26, -37, 25, -8, 1;
-141, 210, -155, 60, -12, 1;
1034, -1575, 1215, -516, 126, -17, 1;
-9693, 14943, -11806, 5270, -1426, 238, -23, 1;
111522, -173109, 138660, -63696, 18267, -3417, 414, -30, 1;
-1528112, 2381814, -1923765, 899226, -267084, 53431, -7337, 675, -38, 1; ...

Paul D. Hanna, Rows n=0..45, as a table of n, a(n) for n=0..1080.

Cf. A098568, A107876; unsigned columns: A107881, A107886.

/* Matrix Inverse of A122176 */
{T(n,k)=local(M=matrix(n+1,n+1,r,c,if(r>=c,binomial((c-1)*(c-2)/2+r,r-c)))); return((M^-1)[n+1,k+1])}
for(n=0,10,for(k=0,n,print1(T(n,k),", "));print(""))

/* Obtain by G.F. */
{T(n,k)=polcoeff(1-sum(j=0, n-k-1, T(j+k,k)*x^j/(1-x+x*O(x^n))^(j*(j+1)/2+j*k+k*(k+1)/2+2)), n-k)}
for(n=0,10,for(k=0,n,print1(T(n,k),", "));print(""))

(1) T(n,k) = A121435(n-1,k) - A121435(n-1,k+1).
(2) T(n,k) = (-1)^(n-k)*[A107876^(k*(k+1)/2 + 2)](n,k);
i.e., column k equals signed column k of A107876^(k*(k+1)/2 + 2).
G.f.s for column k:
(3) 1 = Sum_{j>=0} T(j+k,k)*x^j/(1-x)^( j*(j+1)/2) + j*k + k*(k+1)/2 + 2);
(4) 1 = Sum_{j>=0} T(j+k,k)*x^j*(1+x)^( j*(j-1)/2) + j*k + k*(k+1)/2 + 2).

A121435 Matrix inverse of triangle A122175, where A122175(n,k) = C( k*(k+1)/2 + n-k, n-k) for n>=k>=0.

Original entry on oeis.org

1, -1, 1, 1, -2, 1, -2, 5, -4, 1, 7, -19, 18, -7, 1, -37, 104, -106, 49, -11, 1, 268, -766, 809, -406, 110, -16, 1, -2496, 7197, -7746, 4060, -1210, 216, -22, 1, 28612, -82910, 90199, -48461, 15235, -3032, 385, -29, 1, -391189, 1136923, -1244891, 678874, -220352, 46732, -6699, 638, -37, 1

Offset: 0

Paul D. Hanna, Aug 27 2006

			Triangle begins:
1;
-1, 1;
1, -2, 1;
-2, 5, -4, 1;
7, -19, 18, -7, 1;
-37, 104, -106, 49, -11, 1;
268, -766, 809, -406, 110, -16, 1;
-2496, 7197, -7746, 4060, -1210, 216, -22, 1;
28612, -82910, 90199, -48461, 15235, -3032, 385, -29, 1;
-391189, 1136923, -1244891, 678874, -220352, 46732, -6699, 638, -37, 1; ...

Cf. A098568, A107876; unsigned columns: A107877, A107882.

/* Matrix Inverse of A122175 */ T(n,k)=local(M=matrix(n+1,n+1,r,c,if(r>=c,binomial((c-1)*(c-2)/2+r-1,r-c)))); return((M^-1)[n+1,k+1])

/* Obtain by G.F. */ T(n,k)=polcoeff(1-sum(j=0, n-k-1, T(j+k,k)*x^j/(1-x+x*O(x^n))^(j*(j+1)/2+j*k+k*(k+1)/2+1)), n-k)

(1) T(n,k) = A121434(n-1,k) - A121434(n-1,k+1).
(2) T(n,k) = (-1)^(n-k)*[A107876^(k*(k+1)/2 + 1)](n,k); i.e., column k equals signed column k of matrix power A107876^(k*(k+1)/2 + 1).
G.f.s for column k:
(3) 1 = Sum_{j>=0} T(j+k,k)*x^j/(1-x)^( j*(j+1)/2) + j*k + k*(k+1)/2 + 1);
(4) 1 = Sum_{j>=0} T(j+k,k)*x^j*(1+x)^( j*(j-1)/2) + j*k + k*(k+1)/2 + 1).

cp's OEIS Frontend

A098569 Row sums of the triangle of triangular binomial coefficients given by A098568.

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A121434 Matrix inverse of triangle A098568, where A098568(n, k) = C( (k+1)*(k+2)/2 + n-k-1, n-k) for n>=k>=0.

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A131338 Triangle, read by rows of n*(n+1)/2 + 1 terms, that starts with a '1' in row 0 with row n consisting of n '1's followed by the partial sums of the prior row.

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A290428 Array read by antidiagonals: T(n,k) is the number of graphs with n edges and k vertices, allowing loops and multi-edges.

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A122176 Triangle, read by rows, where T(n,k) = C( k*(k+1)/2 + n-k + 1, n-k) for n>=k>=0.

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A122177 Triangle, read by rows, where T(n,k) = C( k*(k+1)/2 + n-k + 2, n-k) for n>=k>=0.

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A122175 Triangle, read by rows, where T(n,k) = C( k*(k+1)/2 + n-k, n-k) for n>=k>=0.

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A122178 Triangle, read by rows, where T(n,k) = C( n*(n+1)/2 + n-k - 1, n-k), for n>=k>=0.

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