A107876 -id:A107876 - cp's OEIS frontend

A107870 Triangle, read by rows, where T(n,k) = C(n(n-1)/2-k(k-1)/2+n-k+2, n-k).

1, 3, 1, 10, 4, 1, 56, 21, 5, 1, 495, 165, 36, 6, 1, 6188, 1820, 364, 55, 7, 1, 100947, 26334, 4845, 680, 78, 8, 1, 2035800, 475020, 80730, 10626, 1140, 105, 9, 1, 48903492, 10295472, 1623160, 201376, 20475, 1771, 136, 10, 1, 1362649145, 260932815

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;
3,1;
10,4,1;
56,21,5,1;
495,165,36,6,1;
6188,1820,364,55,7,1;
100947,26334,4845,680,78,8,1;
2035800,475020,80730,10626,1140,105,9,1; ...

Cf. A107862, A107865, A107867, A107876, A107871, A107872.

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

A107873 Triangle, read by rows, where T(n,k) = binomial(n(n-1)/2 - k(k-1)/2 + n-k+3, n-k).

Original entry on oeis.org

1, 4, 1, 15, 5, 1, 84, 28, 6, 1, 715, 220, 45, 7, 1, 8568, 2380, 455, 66, 8, 1, 134596, 33649, 5985, 816, 91, 9, 1, 2629575, 593775, 98280, 12650, 1330, 120, 10, 1, 61523748, 12620256, 1947792, 237336, 23751, 2024, 153, 11, 1, 1677106640, 314457495, 45379620, 5245786, 501942, 40920, 2925, 190, 12, 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;
        4,      1;
       15,      5,     1;
       84,     28,     6,     1;
      715,    220,    45,     7,    1;
     8568,   2380,   455,    66,    8,   1;
   134596,  33649,  5985,   816,   91,   9,  1;
  2629575, 593775, 98280, 12650, 1330, 120, 10, 1; ...

Harvey P. Dale, Table of n, a(n) for n = 0..5000

Cf. A107862, A107865, A107867, A107870, A107874, A107875, A107876.

[Binomial(3+Floor((n-k)*(n+k+1)/2), n-k): k in [0..n], n in [0..12]]; // G. C. Greubel, Feb 19 2022

Flatten[Table[Binomial[(n(n-1))/2-(k(k-1))/2+n-k+3,n-k],{n,0,10},{k,0,n}]] (* Harvey P. Dale, Oct 03 2015 *)

T(n,k)=binomial(n*(n-1)/2-k*(k-1)/2 +n-k+3,n-k)

flatten([[binomial(3+(n-k)*(n+k+1)/2, n-k) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Feb 19 2022

From G. C. Greubel, Feb 19 2022: (Start)
T(n,k) = binomial(n*(n-1)/2 - k*(k-1)/2 + n-k+3, n-k).
T(n, 0) = A107874(n).
T(n, 1) = A107875(n). (End)

A126460 Triangle T, read by rows, where column k of matrix power T^( k(k+1)/2 ) equals left-shifted column (k-1) of T for k>=1.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 3, 3, 1, 1, 21, 21, 6, 1, 1, 274, 274, 75, 10, 1, 1, 5806, 5806, 1565, 195, 15, 1, 1, 182766, 182766, 48950, 5940, 420, 21, 1, 1, 8034916, 8034916, 2145626, 257300, 17570, 798, 28, 1, 1, 471517614, 471517614, 125727238, 14989472, 1006880

Offset: 0

Paul D. Hanna, Dec 27 2006

Amazingly, A126460 = A126445^-1*A126450 = A126450^-1*A126454 = A126454^-1*A126457; and also A126465 = A126450*A126445^-1 = A126454*A126450^-1 = A126457*A126454^-1. Also, column k equals unsigned column k of the matrix inverse of triangle P_k defined by P_k(m,j) = C( C(j+2,3) - C(k+2,3) + m-j, m-j) for m>=j>=0.

			Triangle T begins:
1;
1, 1;
1, 1, 1;
3, 3, 1, 1;
21, 21, 6, 1, 1;
274, 274, 75, 10, 1, 1;
5806, 5806, 1565, 195, 15, 1, 1;
182766, 182766, 48950, 5940, 420, 21, 1, 1;
8034916, 8034916, 2145626, 257300, 17570, 798, 28, 1, 1; ...
where column 1 of T^1 equals left-shifted column 0 of T.
Matrix cube T^3 begins:
1;
3, 1;
6, 3, (1);
22, 12, (3), 1;
163, 91, (21), 3, 1;
2167, 1219, (274), 33, 3, 1;
46248, 26091, (5806), 661, 48, 3, 1;
1460301, 824853, (182766), 20341, 1369, 66, 3, 1; ...
where column 2 of T^3 equals left-shifted column 1 of T.
Matrix power T^6 begins:
1;
6, 1;
21, 6, 1;
98, 33, 6, (1);
791, 281, 51, (6), 1;
10850, 3929, 710, (75), 6, 1;
234472, 85557, 15425, (1565), 105, 6, 1;
7444172, 2725402, 490806, (48950), 3080, 141, 6, 1; ...
where column 3 of T^6 equals left-shifted column 2 of T.

Columns: A126461, A126462, A126463, A126464; A126465 (dual); A107876 (variant); subpartitions defined: A115728.

{T(n,k)=abs((matrix(n+1,n+1,r,c, binomial((c-1)*c*(c+1)/3!-k*(k+1)*(k+2)/3!+r-c,r-c))^-1)[n+1,k+1])}
for(n=0,10,for(k=0,n,print1(T(n,k),", "));print(""))

/* As Defined by Matrix Product A126460 = A126445^-1*A126450: */
{T(n,k)=local(M=matrix(n+1,n+1,r,c,if(r>=c,binomial((r-1)*r*(r+1)/3!-(c-1)*c*(c+1)/3!,r-c))), N=matrix(n+1,n+1,r,c,if(r>=c,binomial((r-1)*r*(r+1)/3!-(c-1)*c*(c+1)/3!+1,r-c)))); (M^-1*N)[n+1,k+1]}
for(n=0,10,for(k=0,n,print1(T(n,k),", "));print(""))

G.f. of column k: 1/(1-x) = Sum_{n>=0} T(n+k,k)*x^n*(1-x)^p_k(n), so that column k equals the number of subpartitions of the partition p_k defined by: p_k(n) = (n^2 + (3*k+3)*n + (3*k^2+6*k-4))*n/6 for n>=0.

A107865 Matrix inverse of triangle A107862.

Original entry on oeis.org

1, -1, 1, -1, -2, 1, -7, -4, -3, 1, -77, -26, -9, -4, 1, -1145, -287, -67, -16, -5, 1, -21410, -4412, -798, -139, -25, -6, 1, -481683, -86004, -13029, -1830, -251, -36, -7, 1, -12655196, -2017658, -268368, -32191, -3667, -412, -49, -8, 1, -379998938, -55134458, -6630228, -705680, -69868, -6657, -631

Offset: 0

Paul D. Hanna, Jun 04 2005

Matrix product with A107867 equals A107876.

			Triangle begins:
1;
-1,1;
-1,-2,1;
-7,-4,-3,1;
-77,-26,-9,-4,1;
-1145,-287,-67,-16,-5,1;
-21410,-4412,-798,-139,-25,-6,1;
-481683,-86004,-13029,-1830,-251,-36,-7,1; ...

Cf. A107862, A107866 (column 0), A107867, A107870, A107876.

{T(n,k)=(matrix(n+1,n+1,r,c,if(r>=c, binomial((r-1)*(r-2)/2-(c-1)*(c-2)/2+r-c,r-c)))^-1)[n+1,k+1]}

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
				
```

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).

A141760 Triangle T, read by rows, where the g.f. of column k in matrix power T^m is given by: 1/(1-x)^m = Sum_{n>=k} [T^m](n,k) * x^(n-k)/(1+x)^{n(n-1)/2 - k(k-1)/2} for k>=0.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 6, 6, 3, 1, 1, 26, 26, 13, 4, 1, 1, 154, 154, 77, 23, 5, 1, 1, 1188, 1188, 594, 175, 36, 6, 1, 1, 11474, 11474, 5737, 1678, 336, 52, 7, 1, 1, 134432, 134432, 67216, 19579, 3863, 576, 71, 8, 1, 1, 1863168, 1863168, 931584, 270683, 52944, 7731

Offset: 0

Paul D. Hanna, Jul 18 2008

			Triangle T begins:
1;
1, 1;
1, 1, 1;
2, 2, 1, 1;
6, 6, 3, 1, 1;
26, 26, 13, 4, 1, 1;
154, 154, 77, 23, 5, 1, 1;
1188, 1188, 594, 175, 36, 6, 1, 1;
11474, 11474, 5737, 1678, 336, 52, 7, 1, 1;
134432, 134432, 67216, 19579, 3863, 576, 71, 8, 1, 1;
1863168, 1863168, 931584, 270683, 52944, 7731, 911, 93, 9, 1, 1; ...
Matrix square, T^2, begins:
1;
2, 1;
3, 2, 1;
7, 5, 2, 1;
23, 17, 7, 2, 1;
105, 79, 33, 9, 2, 1;
641, 487, 205, 55, 11, 2, 1;
5034, 3846, 1626, 433, 83, 13, 2, 1; ...
where g.f. for column k of matrix square T^2 is:
1/(1-x)^2 = Sum_{n>=0} [T^2](n,k)*x^(n-k)/(1+x)^{n(n-1)/2 - k(k-1)/2}.
Matrix inverse, T^-1, begins:
1;
-1, 1;
0, -1, 1;
0, -1, -1, 1;
0, -2, -2, -1, 1;
0, -7, -7, -3, -1, 1;
0, -37, -37, -15, -4, -1, 1;
0, -268, -268, -106, -26, -5, -1, 1; ...
Let U = unsigned T^-1 with leftmost column dropped,
then U = A107876 where [U^k](n,k) = U(n,k-1) for n>=k>0.
The g.f. for column k of matrix inverse T^-1 is:
1-x = Sum_{n>=0} [T^-1](n,k) * x^(n-k)/(1+x)^{n(n-1)/2 - k(k-1)/2}.
MATRIX PRODUCTS:
T = P(1)^-1 * P(2) = P(2)^-1 * P(3) = P(m)^-1 * P(m+1);
P(1) begins:
1;
1, 1;
2, 2, 1;
8, 7, 3, 1;
57, 42, 16, 4, 1;
638, 386, 130, 29, 5, 1;
9949, 4944, 1471, 299, 46, 6, 1; ...
where [P(1)](n,k) = [x^(n-k)] 1/(1-x)*(1+x)^{n(n-1)/2-k(k-1)/2};
P(2) begins:
1;
2, 1;
5, 3, 1;
20, 12, 4, 1;
129, 72, 23, 5, 1;
1268, 630, 187, 38, 6, 1;
17548, 7599, 2063, 392, 57, 7, 1; ...
where [P(2)](n,k) = [x^(n-k)] 1/(1-x)^2*(1+x)^{n(n-1)/2-k(k-1)/2};
P(3) begins:
1;
3, 1;
9, 4, 1;
38, 18, 5, 1;
240, 111, 31, 6, 1;
2223, 955, 256, 48, 7, 1;
28672, 11124, 2794, 500, 69, 8, 1; ...
where [P(3)](n,k) = [x^(n-k)] 1/(1-x)^3*(1+x)^{n(n-1)/2-k(k-1)/2}.

Cf. columns: A141761, A141762, A141763; A107876 (unsigned inverse).

T[n_, k_, m_] := T[n, k, m] = If[nJean-François Alcover, Sep 19 2016, adapted from PARI *)

PARI
```
T(n,k,m=1)=if(n
				
```

Matrix powers satisfy: T^m = P(i)^-1 * P(m+i) for all m and i, where P(m) is given by:
[P(m)](n,k) = [x^(n-k)] 1/(1-x)^m * (1+x)^{n(n-1)/2 - k(k-1)/2} for n>=k>=0.
Let U = unsigned matrix inverse (T^-1) with leftmost column dropped, then U = A107876 where [U^k](n,k) = U(n,k-1) for n>=k>0.
G.f. for column k of T: 1/(1-x) = Sum_{n>=0} T(n,k)*x^(n-k)/(1+x)^{n(n-1)/2 - k(k-1)/2}.
T(n,k) = 1 - Sum_{j=k..n-1} T(j,k)*(-1)^(n-j)*C(j(j-1)/2 - k(k-1)/2 + n-j-1, n-j) for n>k with T(k,k)=1 for k>=0.
G.f. for column k of matrix power T^m:
1/(1-x)^m = Sum_{n>=0} [T^m](n,k)*x^(n-k)/(1+x)^{n*(n-1)/2 - k*(k-1)/2}.
[T^m](n,k) = C(m+n-1,n) - Sum_{j=k..n-1} [T^m](j,k)*(-1)^(n-j)*C(j(j-1)/2 - k(k-1)/2 + n-j-1,n-j) for n>k with [T^m](k,k)=1 for k>=0.

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).

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A107870 Triangle, read by rows, where T(n,k) = C(n*(n-1)/2-k*(k-1)/2+n-k+2, n-k).

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A107873 Triangle, read by rows, where T(n,k) = binomial(n*(n-1)/2 - k*(k-1)/2 + n-k+3, n-k).

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A126460 Triangle T, read by rows, where column k of matrix power T^( k(k+1)/2 ) equals left-shifted column (k-1) of T for k>=1.

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A107865 Matrix inverse of triangle A107862.

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

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A107870 Triangle, read by rows, where T(n,k) = C(n(n-1)/2-k(k-1)/2+n-k+2, n-k).

A107873 Triangle, read by rows, where T(n,k) = binomial(n(n-1)/2 - k(k-1)/2 + n-k+3, n-k).