A107876 -id:A107876 - cp's OEIS frontend

A107877 Column 1 of triangle A107876.

1, 1, 2, 7, 37, 268, 2496, 28612, 391189, 6230646, 113521387, 2332049710, 53384167192, 1348601249480, 37291381915789, 1120914133433121, 36406578669907180, 1271084987848923282, 47487293697623885913, 1890771531272515677250, 79947079338974990793060

Offset: 0

Paul D. Hanna, Jun 04 2005, Apr 10 2007

nonn

Also number of subpartitions of partition consisting of first n-1 triangular numbers; e.g., a(4) = subp([1,3,6]) = 37. - Franklin T. Adams-Watters, Jun 26 2006
Number of length-n restricted growth strings (RGS) [s(0),s(1),...,s(n-1)] where s(0)=0 and s(k) <= s(k-1)+k, see Fxtbook link and example. - Joerg Arndt, Apr 30 2011
Number of Dyck paths whose ascent lengths are exactly {1,2,...,n+1}; for example, the a(2) = 2 paths are uduuduuudddd and uduudduuuddd. - David Scambler, May 30 2012
Number of types of cells of a fine mixed subdivision of the Tesler flow polytope. - Alejandro H. Morales, Oct 11 2017

			1 = 1*(1-x)^1 + 1*x*(1-x)^2 + 2*x^2*(1-x)^4 + 7*x^3*(1-x)^7 + 37*x^4*(1-x)^11 + 268*x^5*(1-x)^16 + 2496*x^6*(1-x)^22 + ...
Also equals the final term in rows of the triangle where row n+1 equals the partial sums of row n with the final term repeated n+1 times, starting with a '1' in row 0, as illustrated by:
1;
1, 1;
1, 2,  2,  2;
1, 3,  5,  7,  7,  7,   7;
1, 4,  9, 16, 23, 30,  37,  37,  37,  37,  37;
1, 5, 14, 30, 53, 83, 120, 157, 194, 231, 268, 268, 268, 268, 268, 268; ...
Restricted growth strings: a(0)=1 corresponds to the empty string; a(1)=1 to [0];
a(2) = 2 to [00] and [01]; a(3)=7 to
  1:  [ 0 0 0 ],
  2:  [ 0 0 1 ],
  3:  [ 0 0 2 ],
  4:  [ 0 1 0 ],
  5:  [ 0 1 1 ],
  6:  [ 0 1 2 ],
  7:  [ 0 1 3 ].
[_Joerg Arndt_, Apr 30 2011]

R. P. Stanley, Enumerative Combinatorics volume 1, 2nd edition, Cambridge University Press, 2011, Ch. 3

Alois P. Heinz, Table of n, a(n) for n = 0..390
Joerg Arndt, Matters Computational (The Fxtbook), section 17.3.6, pp. 368-369
K. Mészáros, A. H. Morales, Volumes and Ehrhart polynomials of flow polytopes, arXiv:1710.00701 [math.CO], 2017, sections 6.1 and 7.

Cf. A107876, A107878, A107879, A115728, A115729.

Cf. A305605, A305601.

b:= proc(n, y) option remember; `if`(n=0, 1, add(
      b(n-1, y+i-n), i=max(1, n-y)..n*(n-1)/2+1-y))
    end:
a:= n-> b(n+1, 0):
seq(a(n), n=0..25);  # Alois P. Heinz, Nov 26 2016
# second Maple program:
a:= n-> LinearAlgebra:-Determinant(Matrix(n,(i,j)->
        binomial(binomial(n+1-i,2)+1,i-j+1))):
seq(a(n), n=0..25); # Alejandro H. Morales, Aug 31 2017

a[ n_, k_: 1, j_: 1] := If[ n < 2, Boole[n >= 0], a[n, k, j] = Sum[a[n - 1, i, j + 1], {i, k + j}]]; (* Michael Somos, Nov 26 2016 *)

{a(n)=polcoeff(1-sum(k=0,n-1,a(k)*x^k*(1-x+x*O(x^n))^(1+k*(k+1)/2)),n)}

G.f.: 1 = Sum_{k>=0} a(k)*x^k*(1-x)^(1 + k*(k+1)/2).
G.f.: 1 = Sum_{k>=0} a(k)*x^k/(1+x)^((k+1)*(k+2)/2).
From Benedict W. J. Irwin, Nov 26 2016: (Start)
Conjecture: a(n) can be expressed with a series of nested sums,
a(3) = Sum_{i=1..2} i+2,
a(4) = Sum_{i=1..2} Sum_{j=1..i+2} j+3,
a(5) = Sum_{i=1..2} Sum_{j=1..i+2} Sum_{k=1..j+3} k+4,
a(6) = Sum_{i=1..2} Sum_{j=1..i+2} Sum_{k=1..j+3} Sum_{l=1..k+4} l+5. (End)
Determinantal formula: a(n) = Det(A) where A is the n X n matrix with entries A(i,j) = binomial(binomial(n+1-i,2)+1,i-j+1). This follows by the formula by MacMahon (see EC1 Ex 3.63) for the number of such subpartitions. -  Alejandro H. Morales, Aug 31 2017

A107884 Matrix cube of triangle A107876; equals the product of triangular matrices: A107876^3 = A107862^-1*A107873.

Original entry on oeis.org

1, 3, 1, 6, 3, 1, 16, 9, 3, 1, 63, 37, 12, 3, 1, 351, 210, 67, 15, 3, 1, 2609, 1575, 498, 106, 18, 3, 1, 24636, 14943, 4701, 975, 154, 21, 3, 1, 284631, 173109, 54298, 11100, 1689, 211, 24, 3, 1, 3909926, 2381814, 745734, 151148, 22518, 2688, 277, 27, 3, 1

Offset: 0

Paul D. Hanna, Jun 04 2005

Column 0 is A107885.
Column 1 is A107886.
Column 2 equals A107887.
Column 3 equals SHIFT_LEFT(A107878), where A107878 is column 2 of A107876.
Column 4 equals A107888.

			G.f. for column 0:
1 = T(0,0)*(1-x)^3 + T(1,0)*x*(1-x)^3 + T(2,0)*x^2*(1-x)^4 + T(3,0)*x^3*(1-x)^6 + T(4,0)*x^4*(1-x)^9 + T(5,0)*x^5*(1-x)^13 + ...
  = 1*(1-x)^3 + 3*x*(1-x)^3 + 6*x^2*(1-x)^4 + 16*x^3*(1-x)^6 + 63*x^4*(1-x)^9 + 351*x^5*(1-x)^13 + ...
G.f. for column 1:
1 = T(1,1)*(1-x)^3 + T(2,1)*x*(1-x)^4 + T(3,1)*x^2*(1-x)^6 + T(4,1)*x^3*(1-x)^9 + T(5,1)*x^4*(1-x)^13 + T(6,1)*x^5*(1-x)^18 + ...
  = 1*(1-x)^3 + 3*x*(1-x)^4 + 9*x^2*(1-x)^6 + 37*x^3*(1-x)^9 + 210*x^4*(1-x)^13 + 1575*x^5*(1-x)^18 + ...
Triangle begins:
       1;
       3,      1;
       6,      3,     1;
      16,      9,     3,     1;
      63,     37,    12,     3,    1;
     351,    210,    67,    15,    3,   1;
    2609,   1575,   498,   106,   18,   3,  1;
   24636,  14943,  4701,   975,  154,  21,  3, 1;
  284631, 173109, 54298, 11100, 1689, 211, 24, 3, 1;
  ...

Cf. A107862, A107870, A107873, A107867, A107876, A107880, A107884, A107885, A107886, A107887, A107888.

max = 10;
A107862 = Table[Binomial[If[n < k, 0, n*(n-1)/2-k*(k-1)/2 + n - k], n - k], {n, 0, max}, {k, 0, max}];
A107867 = Table[Binomial[If[n < k, 0, n*(n-1)/2-k*(k-1)/2 + n-k+1], n - k], {n, 0, max}, {k, 0, max}];
T = MatrixPower[Inverse[A107862].A107867, 3];
Table[T[[n+1, k+1]], {n, 0, max}, {k, 0, n}] // Flatten (* Jean-François Alcover, May 31 2024 *)

{T(n,k)=polcoeff(1-sum(j=0,n-k-1, T(j+k,k)*x^j*(1-x+x*O(x^n))^(3+(k+j)*(k+j-1)/2-k*(k-1)/2)),n-k)}

G.f. for column k: 1 = Sum_{j>=0} T(k+j, k)*x^j*(1-x)^(3 + (k+j)*(k+j-1)/2 - k*(k-1)/2).

A107878 Column 2 of triangle A107876.

Original entry on oeis.org

1, 1, 3, 15, 106, 975, 11100, 151148, 2401365, 43681578, 896371205, 20504034645, 517705752096, 14310162565395, 430020328711305, 13963933247986995, 487456219774434795, 18209055555140970945, 724952705958984299025, 30650849492427960893946, 1371796147488157950190065

Offset: 0

Paul D. Hanna, Jun 04 2005

nonn

			G.f. = 1 + x + 3*x^2 + 15*x^3 + 106*x^4 + 975*x^5 + 11100*x^6 + 151148*x^7 + ...
1 = 1*(1-x)^1 + 1*x*(1-x)^3 + 3*x^2*(1-x)^6 + 15*x^3*(1-x)^10 + 106*x^4*(1-x)^15 + 975*x^5*(1-x)^21 + 11100*x^6*(1-x)^21 +...

Alois P. Heinz, Table of n, a(n) for n = 0..390

Cf. A000217, A107876, A107877, A107879, A209440 (variant).

a:= proc(n) option remember; `if`(n=0, 1, add(a(j-1)*
      (-1)^(n-j)*binomial(j*(j+1)/2, n-j+1), j=1..n))
    end:
seq(a(n), n=0..22);  # Alois P. Heinz, Jul 10 2022

a[ n_, k_: 0, j_: 1] := If[ n < 1, Boole[n >= 0], a[ n, k, j] = Sum[ a[ n - 1, i, j + 1], {i, k + j}]]; (* Michael Somos, Nov 26 2016 *)

{a(n)=polcoeff(1-sum(k=0,n-1,a(k)*x^k*(1-x+x*O(x^n))^((k+1)*(k+2)/2)),n)}

G.f.: 1 = Sum_{k>=0} a(k)*x^k*(1-x)^((k+1)*(k+2)/2).
From Benedict W. J. Irwin, Nov 29 2016: (Start)
Conjecture: a(n) is given by a series of nested sums,
a(1) = Sum_{i=1..1} 1,
a(2) = Sum_{i=1..1} Sum_{j=1..i+2} 1,
a(3) = Sum_{i=1..1} Sum_{j=1..i+2} Sum_{k=1..j+3} 1,
a(4) = Sum_{i=1..1} Sum_{j=1..i+2} Sum_{k=1..j+3} Sum_{l=1..k+4} 1. (End)

A107880 Matrix square of triangle A107876; equals matrix product of triangles: A107876^2 = A107862^-1A107870 = A107867^-1A107873.

Original entry on oeis.org

1, 2, 1, 3, 2, 1, 7, 5, 2, 1, 26, 19, 7, 2, 1, 141, 104, 37, 9, 2, 1, 1034, 766, 268, 61, 11, 2, 1, 9693, 7197, 2496, 550, 91, 13, 2, 1, 111522, 82910, 28612, 6195, 982, 127, 15, 2, 1, 1528112, 1136923, 391189, 83837, 12977, 1596, 169, 17, 2, 1, 24372513, 18141867, 6230646, 1326923, 202494, 24206, 2424, 217, 19, 2, 1

Offset: 0

Paul D. Hanna, Jun 04 2005

Column 0 is A107881. Column 1 is A107882. Column 3 equals A107883. Column 2 equals SHIFT_LEFT(A107877), where A107877 is column 1 of A107876.

			G.f. for column 0:
1 = T(0,0)*(1-x)^2 + T(1,0)*x*(1-x)^2 + T(2,0)*x^2*(1-x)^3 + T(3,0)*x^3*(1-x)^5 + T(4,0)*x^4*(1-x)^8 + T(5,0)*x^5*(1-x)^12 +...
= 1*(1-x)^2 + 2*x*(1-x)^2 + 3*x^2*(1-x)^3 + 7*x^3*(1-x)^5 + 26*x^4*(1-x)^8 + 141*x^5*(1-x)^12 +...
G.f. for column 1:
1 = T(1,1)*(1-x)^2 + T(2,1)*x*(1-x)^3 + T(3,1)*x^2*(1-x)^5 + T(4,1)*x^3*(1-x)^8 + T(5,1)*x^4*(1-x)^12 + T(6,1)*x^5*(1-x)^17 +...
= 1*(1-x)^2 + 2*x*(1-x)^3 + 5*x^2*(1-x)^5 + 19*x^3*(1-x)^8 + 104*x^4*(1-x)^12 + 766*x^5*(1-x)^17 +...
Triangle T begins:
  1;
  2,1;
  3,2,1;
  7,5,2,1;
  26,19,7,2,1;
  141,104,37,9,2,1;
  1034,766,268,61,11,2,1;
  9693,7197,2496,550,91,13,2,1;
  111522,82910,28612,6195,982,127,15,2,1;
  ...

Cf. A107862, A107870, A107873, A107867, A107876, A107884, A107887.

max = 10;
A107862 = Table[Binomial[If[n < k, 0, n*(n-1)/2-k*(k-1)/2 + n - k], n - k], {n, 0, max}, {k, 0, max}];
A107867 = Table[Binomial[If[n < k, 0, n*(n-1)/2-k*(k-1)/2 + n-k+1], n - k], {n, 0, max}, {k, 0, max}];
T = MatrixPower[Inverse[A107862].A107867, 2];
Table[T[[n+1, k+1]], {n, 0, max}, {k, 0, n}] // Flatten (* Jean-François Alcover, May 31 2024 *)

{T(n,k)=polcoeff(1-sum(j=0,n-k-1, T(j+k,k)*x^j*(1-x+x*O(x^n))^(2+(k+j)*(k+j-1)/2-k*(k-1)/2)),n-k)}

G.f. for column k: 1 = Sum_{j>=0} T(k+j, k)*x^j*(1-x)^(2+(k+j)*(k+j-1)/2-k*(k-1)/2).

A107879 Column 3 of triangle A107876.

Original entry on oeis.org

1, 1, 4, 26, 230, 2565, 34516, 544423, 9857583, 201664780, 4603336725, 116059191472, 3204682702923, 96226940232235, 3122975927539860, 108970956192622980, 4069312064491308140, 161969813446983961395, 6846708764857861662741, 306381118887919045527510

Offset: 0

Paul D. Hanna, Jun 04 2005

nonn

			1 = 1*(1-x)^1 + 1*x*(1-x)^4 + 4*x^2*(1-x)^8 + 26*x^3*(1-x)^13 + 230*x^4*(1-x)^19 + 2565*x^5*(1-x)^26 + 34516*x^6*(1-x)^34 +...

Alois P. Heinz, Table of n, a(n) for n = 0..389

Cf. A107876, A107877, A107878.

a:= proc(n) option remember; `if`(n=0, 1, add(a(j-1)*
      (-1)^(n-j)*binomial((j+1)*(j+2)/2-2, n-j+1), j=1..n))
    end:
seq(a(n), n=0..22);  # Alois P. Heinz, Jul 10 2022

{a(n)=polcoeff(1-sum(k=0,n-1,a(k)*x^k*(1-x+x*O(x^n))^((k+2)*(k+3)/2-2)),n)}

G.f.: 1 = Sum_{k>=0} a(k)*x^k*(1-x)^((k+2)*(k+3)/2 - 2).

A107889 Triangular matrix T, read by rows, that satisfies: [T^-k](n,k) = -T(n,k-1) for n >= k > 0, or, equivalently, (column k of T^-k) = -SHIFT_LEFT(column k-1 of T) when zeros above the diagonal are ignored. Also, matrix inverse of triangle A107876.

Original entry on oeis.org

1, -1, 1, 0, -1, 1, 0, -1, -1, 1, 0, -3, -2, -1, 1, 0, -15, -9, -3, -1, 1, 0, -106, -61, -18, -4, -1, 1, 0, -975, -550, -154, -30, -5, -1, 1, 0, -11100, -6195, -1689, -310, -45, -6, -1, 1, 0, -151148, -83837, -22518, -4005, -545, -63, -7, -1, 1, 0, -2401365, -1326923, -353211, -61686, -8105, -875, -84, -8, -1, 1

Offset: 0

Paul D. Hanna, Jun 05 2005

SHIFT_LEFT(column 1) = -A107878.
SHIFT_LEFT(column 2) = -A107883.
SHIFT_LEFT(column 3) = -A107888.

			G.f. for column 1:
1 = T(1,1)*(1-x)^-1 + T(2,1)*x*(1-x)^0 + T(3,1)*x^2*(1-x)^2 + T(4,1)*x^3*(1-x)^5 + T(5,1)*x^4*(1-x)^9 + T(6,1)*x^5*(1-x)^14 + ...
  = 1*(1-x)^-1 - 1*x*(1-x)^0 - 1*x^2*(1-x)^2 - 3*x^3*(1-x)^5 - 15*x^4*(1-x)^9 - 106*x^5*(1-x)^14 - 975*x^6*(1-x)^20 + ...
G.f. for column 2:
1 = T(2,2)*(1-x)^-1 + T(3,2)*x*(1-x)^1 + T(4,2)*x^2*(1-x)^4 + T(5,2)*x^3*(1-x)^8 + T(6,2)*x^4*(1-x)^13 + T(7,2)*x^5*(1-x)^19 + ...
  = 1*(1-x)^-1 - 1*x*(1-x)^1 - 2*x^2*(1-x)^4 - 9*x^3*(1-x)^8 - 61*x^4*(1-x)^13 - 550*x^5*(1-x)^19 - 6195*x^6*(1-x)^26 + ...
Triangle begins:
   1;
  -1,      1;
   0,     -1,     1;
   0,     -1,    -1,     1;
   0,     -3,    -2,    -1,    1;
   0,    -15,    -9,    -3,   -1,   1;
   0,   -106,   -61,   -18,   -4,  -1,  1;
   0,   -975,  -550,  -154,  -30,  -5, -1,  1;
   0, -11100, -6195, -1689, -310, -45, -6, -1, 1;
  ...

Cf. A107876, A107880, A107884.

max = 10;
A107862 = Table[Binomial[If[n < k, 0, n*(n-1)/2-k*(k-1)/2 + n - k], n - k], {n, 0, max}, {k, 0, max}];
A107867 = Table[Binomial[If[n < k, 0, n*(n-1)/2-k*(k-1)/2 + n-k+1], n - k], {n, 0, max}, {k, 0, max}];
T = Inverse[Inverse[A107862].A107867];
Table[T[[n + 1, k + 1]], {n, 0, max}, {k, 0, n}] // Flatten (* Jean-François Alcover, May 31 2024 *)

{T(n,k)=polcoeff(1-sum(j=0,n-k-1, T(j+k,k)*x^j*(1-x+x*O(x^n))^(-1+(k+j)*(k+j-1)/2-k*(k-1)/2)),n-k)}

G.f. for column k: 1 = Sum_{j>=0} T(k+j, k)*x^j*(1-x)^(-1 + (k+j)*(k+j-1)/2 - k*(k-1)/2).

A101479 Triangular matrix T, read by rows, where row n equals row (n-1) of T^(n-1) after appending '1' for the main diagonal.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 3, 2, 1, 1, 19, 9, 3, 1, 1, 191, 70, 18, 4, 1, 1, 2646, 795, 170, 30, 5, 1, 1, 46737, 11961, 2220, 335, 45, 6, 1, 1, 1003150, 224504, 37149, 4984, 581, 63, 7, 1, 1, 25330125, 5051866, 758814, 92652, 9730, 924, 84, 8, 1, 1, 735180292, 132523155, 18301950, 2065146, 199692, 17226, 1380, 108, 9, 1, 1

Offset: 0

Paul D. Hanna, Jan 21 2005, Jul 26 2006, May 27 2007

Remarkably, T equals the product of these triangular matrices: T = A107867*A107862^-1 = A107870*A107867^-1 = A107873*A107870^-1; reversing the order of these products yields triangle A107876.

			Triangle begins:
1;
1, 1;
1, 1, 1;
3, 2, 1, 1;
19, 9, 3, 1, 1;
191, 70, 18, 4, 1, 1;
2646, 795, 170, 30, 5, 1, 1;
46737, 11961, 2220, 335, 45, 6, 1, 1;
1003150, 224504, 37149, 4984, 581, 63, 7, 1, 1;
25330125, 5051866, 758814, 92652, 9730, 924, 84, 8, 1, 1;
735180292, 132523155, 18301950, 2065146, 199692, 17226, 1380, 108, 9, 1, 1; ...
Row 4 starts with row 3 of T^3 which begins:
1;
3, 1;
6, 3, 1;
19, 9, 3, 1; ...
row 5 starts with row 4 of T^4 which begins:
1;
4, 1;
10, 4, 1;
34, 14, 4, 1;
191, 70, 18, 4, 1; ...
An ALTERNATE GENERATING METHOD is illustrated as follows.
For row 4:
Start with a '1' and append 2 zeros,
take partial sums and append 1 zero,
take partial sums thrice more, resulting in:
1, 0, 0;
1, 1, 1, 0;
1, 2, 3, 3;
1, 3, 6, 9;
1, 4,10,19.
Final nonzero terms form row 4: [19,9,3,1,1].
For row 5:
Start with a '1' and append 3 zeros,
take partial sums and append 2 zeros,
take partial sums and append 1 zero,
take partial sums thrice more, resulting in:
1, 0, 0, 0;
1, 1, 1, 1, 0,  0;
1, 2, 3, 4, 4,  4,  0;
1, 3, 6,10,14, 18, 18;
1, 4,10,20,34, 52, 70;
1, 5,15,35,69,121,191;
where the final nonzero terms form row 5: [191,70,18,4,1,1].
Likewise, for row 6:
1, 0, 0, 0,  0;
1, 1, 1, 1,  1,  0,  0,  0;
1, 2, 3, 4,  5,  5,  5,  5,   0,   0;
1, 3, 6,10, 15, 20, 25, 30,  30,  30,   0;
1, 4,10,20, 35, 55, 80,110, 140, 170, 170;
1, 5,15,35, 70,125,205,315, 455, 625, 795;
1, 6,21,56,126,251,456,771,1226,1851,2646;
where the final nonzero terms form row 6: [2646,795,170,30,5,1,1].
Continuing in this way generates all rows of this triangle.

Alois P. Heinz, Rows n = 0..140, flattened (first 31 rows from Paul D. Hanna)

Columns are A101481, A101482, A101483, row sums form A101484.
Cf. A107876 (dual triangle).
Cf. A304184, A304185, A304186, A304187.

b:= proc(n) option remember;
      Matrix(n, (i,j)-> T(i-1,j-1))^(n-1)
    end:
T:= proc(n,k) option remember;
     `if`(n=k, 1, `if`(k>n, 0, b(n)[n,k+1]))
    end:
seq(seq(T(n,k), k=0..n), n=0..10);  # Alois P. Heinz, Apr 13 2020

b[n_] := b[n] = MatrixPower[Table[T[i-1, j-1], {i, n}, {j, n}], n-1];
T[n_, k_] := T[n, k] = If[n == k, 1, If[k > n, 0, b[n][[n, k+1]]]];
Table[Table[T[n, k], {k, 0, n}], {n, 0, 10}] // Flatten (* Jean-François Alcover, Apr 25 2020, after Alois P. Heinz *)

{T(n,k) = my(A=Mat(1),B); for(m=1,n+1, B=matrix(m,m); for(i=1,m, for(j=1,i, if(j==i,B[i,j]=1, B[i,j] = (A^(i-2))[i-1,j]);)); A=B); return(A[n+1,k+1])}
for(n=0,10, for(k=0,n, print1(T(n,k),", ")); print(""))

{T(n,k) = my(A=vector(n+1),p); A[1]=1; for(j=1,n-k-1, p=(n-1)*(n-2)/2-(n-j-1)*(n-j-2)/2; A = Vec((Polrev(A)+x*O(x^p))/(1-x))); A = Vec((Polrev(A) +x*O(x^p)) / (1-x) ); A[p+1]}
for(n=0,10, for(k=0,n, print1(T(n,k),", ")); print(""))

A107862 Triangle, read by rows, where T(n,k) = C(n(n-1)/2 - k(k-1)/2 + n-k, n-k).

Original entry on oeis.org

1, 1, 1, 3, 2, 1, 20, 10, 3, 1, 210, 84, 21, 4, 1, 3003, 1001, 220, 36, 5, 1, 54264, 15504, 3060, 455, 55, 6, 1, 1184040, 296010, 53130, 7315, 816, 78, 7, 1, 30260340, 6724520, 1107568, 142506, 14950, 1330, 105, 8, 1, 886163135, 177232627, 26978328, 3262623, 324632, 27405, 2024, 136, 9, 1

Offset: 0

Paul D. Hanna, Jun 04 2005

Remarkably, the following matrix products are all equal to A107876: A107862^-1*A107867 = A107867^-1*A107870 = A107870^-1*A107873.

			Triangle begins:
        1;
        1,      1;
        3,      2,     1;
       20,     10,     3,    1;
      210,     84,    21,    4,   1;
     3003,   1001,   220,   36,   5,  1;
    54264,  15504,  3060,  455,  55,  6, 1;
  1184040, 296010, 53130, 7315, 816, 78, 7, 1; ...

G. C. Greubel, Rows n = 0..50 of the triangle, flattened

Cf. A014068 (column 0), A107863 (column 1), A099121 (column 2), A107865, A107867, A107870, A107876.

[Binomial(Floor((n-k)*(n+k+1)/2), n-k): k in [0..n], n in [0..12]]; // G. C. Greubel, Feb 19 2022

T[n_,k_]:= Binomial[(n-k)*(n+k+1)/2, n-k];
Table[T[n,k], {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, Feb 19 2022 *)

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

flatten([[binomial( (n-k)*(n+k+1)/2, n-k) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Feb 19 2022

T(n,k) = binomial( (n-k)*(n+k+1)/2, n-k). - G. C. Greubel, Feb 19 2022

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.

Original entry on oeis.org

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).

A107867 Triangle, read by rows, where T(n,k) = C(n(n-1)/2-k(k-1)/2+n-k+1,n-k).

Original entry on oeis.org

1, 2, 1, 6, 3, 1, 35, 15, 4, 1, 330, 120, 28, 5, 1, 4368, 1365, 286, 45, 6, 1, 74613, 20349, 3876, 560, 66, 7, 1, 1560780, 376740, 65780, 8855, 969, 91, 8, 1, 38608020, 8347680, 1344904, 169911, 17550, 1540, 120, 9, 1, 1101716330, 215553195, 32224114

Offset: 0

Paul D. Hanna, Jun 04 2005

Remarkably, the following matrix products are all equal to A107876: A107862^-1*A107867 = A107867^-1*A107870 = A107870^-1*A107873.

			Triangle begins:
1;
2,1;
6,3,1;
35,15,4,1;
330,120,28,5,1;
4368,1365,286,45,6,1;
74613,20349,3876,560,66,7,1;
1560780,376740,65780,8855,969,91,8,1; ...

Cf. A099121, A107862, A107865, A107867, A107870, A107876.

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

cp's OEIS Frontend

A107877 Column 1 of triangle A107876.

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A107884 Matrix cube of triangle A107876; equals the product of triangular matrices: A107876^3 = A107862^-1*A107873.

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A107878 Column 2 of triangle A107876.

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A107880 Matrix square of triangle A107876; equals matrix product of triangles: A107876^2 = A107862^-1*A107870 = A107867^-1*A107873.

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A107879 Column 3 of triangle A107876.

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A107889 Triangular matrix T, read by rows, that satisfies: [T^-k](n,k) = -T(n,k-1) for n >= k > 0, or, equivalently, (column k of T^-k) = -SHIFT_LEFT(column k-1 of T) when zeros above the diagonal are ignored. Also, matrix inverse of triangle A107876.

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A101479 Triangular matrix T, read by rows, where row n equals row (n-1) of T^(n-1) after appending '1' for the main diagonal.

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A107880 Matrix square of triangle A107876; equals matrix product of triangles: A107876^2 = A107862^-1A107870 = A107867^-1A107873.

A107862 Triangle, read by rows, where T(n,k) = C(n(n-1)/2 - k(k-1)/2 + n-k, n-k).

A107867 Triangle, read by rows, where T(n,k) = C(n(n-1)/2-k(k-1)/2+n-k+1,n-k).