A121434 -id:A121434 - cp's OEIS frontend

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

Paul D. Hanna, Sep 15 2004

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)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

			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)

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.

Cf. A098569. A290428 (unlabeled graphs).
Cf. A121434 (inverse); variants: A122175, A122176, A122177; A107876.
Cf. A131338.

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 *)

{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(""))

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

A121438 Matrix inverse of triangle A122178, where A122178(n,k) = C( n*(n+1)/2 + n-k - 1, n-k) for n>=k>=0.

Original entry on oeis.org

1, -1, 1, -3, -3, 1, -17, -3, -6, 1, -160, -25, 5, -10, 1, -2088, -285, -35, 30, -15, 1, -34307, -4179, -420, -91, 84, -21, 1, -675091, -74823, -6916, -497, -322, 182, -28, 1, -15428619, -1577763, -135639, -10080, -63, -1002, 342, -36, 1, -400928675, -38209725, -3082905, -215700, -14139, 2655, -2625

Offset: 0

Paul D. Hanna, Aug 29 2006

A triangle having similar properties and complementary construction is the dual triangle A121434.

			Triangle, A122178^-1, begins:
1;
-1, 1;
-3, -3, 1;
-17, -3, -6, 1;
-160, -25, 5, -10, 1;
-2088, -285, -35, 30, -15, 1;
-34307, -4179, -420, -91, 84, -21, 1;
-675091, -74823, -6916, -497, -322, 182, -28, 1;
-15428619, -1577763, -135639, -10080, -63, -1002, 342, -36, 1; ...
Triangle A121412 begins:
1;
1, 1;
3, 1, 1;
18, 4, 1, 1;
170, 30, 5, 1, 1; ...
Row 3 of A122178^-1 equals row 3 of A121412^(-6), which begins:
1;
-6, 1;
3, -6, 1;
-17, -3, -6, 1; ...
Row 4 of A122178^-1 equals row 4 of A121412^(-10), which begins:
1;
-10, 1;
25, -10, 1;
-15, 15, -10, 1;
-160, -25, 5, -10, 1; ...

Cf. A122178 (matrix inverse); A121412; variants: A121439, A121440, A121441; A121434 (dual).

/* Matrix Inverse of A122178 */ {T(n,k)=local(M=matrix(n+1,n+1,r,c,if(r>=c,binomial(r*(r-1)/2+r-c-1,r-c)))); return((M^-1)[n+1,k+1])}

T(n,k) = [A121412^(-n*(n+1)/2)](n,k) for n>=k>=0; i.e., row n of A122178^-1 equals row n of matrix power A121412^(-n*(n+1)/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

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.

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A121438 Matrix inverse of triangle A122178, where A122178(n,k) = C( n*(n+1)/2 + n-k - 1, n-k) for n>=k>=0.

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

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