A121335 -id:A121335 - cp's OEIS frontend

A121440 Matrix inverse of triangle A121335, where A121335(n,k) = C( n*(n+1)/2 + n-k + 1, n-k) for n>=k>=0.

1, -3, 1, 0, -5, 1, -12, 4, -8, 1, -129, -22, 18, -12, 1, -1785, -238, -51, 51, -17, 1, -30291, -3634, -345, -161, 115, -23, 1, -608565, -66750, -6111, -285, -505, 225, -30, 1, -14112744, -1432296, -122227, -9177, 665, -1387, 399, -38, 1, -370746528, -35129022, -2818543, -196037, -14335, 4841, -3337, 658

Offset: 0

Paul D. Hanna, Aug 29 2006

A triangle having similar properties and complementary construction is the dual triangle A121436.

			Triangle, A121335^-1, begins:
1;
-3, 1;
0, -5, 1;
-12, 4, -8, 1;
-129, -22, 18, -12, 1;
-1785, -238, -51, 51, -17, 1;
-30291, -3634, -345, -161, 115, -23, 1;
-608565, -66750, -6111, -285, -505, 225, -30, 1;
-14112744, -1432296, -122227, -9177, 665, -1387, 399, -38, 1; ...
Triangle A121412 begins:
1;
1, 1;
3, 1, 1;
18, 4, 1, 1;
170, 30, 5, 1, 1; ...
Row 3 of A121335^-1 equals row 3 of A121412^(-8), which begins:
1;
-8, 1;
12, -8, 1;
-12, 4, -8, 1; ...
Row 4 of A121335^-1 equals row 4 of A121412^(-12), which begins:
1;
-12, 1;
42, -12, 1;
-34, 30, -12, 1;
-129, -22, 18, -12, 1; ...

Cf. A121335 (matrix inverse); A121412; variants: A121438, A121439, A121441; A121436 (dual).

/* Matrix Inverse of A121335 */ {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 - 2)](n,k) for n>=k>=0; i.e., row n of A121335^-1 equals row n of matrix power A121412^(-n*(n+1)/2 - 2).

A121334 Triangle, read by rows, where T(n,k) = C( n*(n+1)/2 + n-k, n-k), for n>=k>=0.

Original entry on oeis.org

1, 2, 1, 10, 4, 1, 84, 28, 7, 1, 1001, 286, 66, 11, 1, 15504, 3876, 816, 136, 16, 1, 296010, 65780, 12650, 2024, 253, 22, 1, 6724520, 1344904, 237336, 35960, 4495, 435, 29, 1, 177232627, 32224114, 5245786, 749398, 91390, 9139, 703, 37, 1, 5317936260

Offset: 0

Paul D. Hanna, Aug 29 2006

A triangle having similar properties and complementary construction is the dual triangle A122175.

			Triangle begins:
1;
2, 1;
10, 4, 1;
84, 28, 7, 1;
1001, 286, 66, 11, 1;
15504, 3876, 816, 136, 16, 1;
296010, 65780, 12650, 2024, 253, 22, 1;
6724520, 1344904, 237336, 35960, 4495, 435, 29, 1;
177232627, 32224114, 5245786, 749398, 91390, 9139, 703, 37, 1; ...

Cf. A121439 (matrix inverse); A121412; variants: A122178, A121335, A121336; A122175 (dual).

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

Remarkably, row n of the matrix inverse (A121439) equals row n of A121412^(-n*(n+1)/2-1). 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'.

A121336 Triangle, read by rows, where T(n,k) = C( n*(n+1)/2 + n-k + 2, n-k), for n>=k>=0.

Original entry on oeis.org

1, 4, 1, 21, 6, 1, 165, 45, 9, 1, 1820, 455, 91, 13, 1, 26334, 5985, 1140, 171, 18, 1, 475020, 98280, 17550, 2600, 300, 24, 1, 10295472, 1947792, 324632, 46376, 5456, 496, 31, 1, 260932815, 45379620, 7059052, 962598, 111930, 10660, 780, 39, 1

Offset: 0

Paul D. Hanna, Aug 29 2006

A triangle having similar properties and complementary construction is the dual triangle A122177.

			Triangle begins:
1;
4, 1;
21, 6, 1;
165, 45, 9, 1;
1820, 455, 91, 13, 1;
26334, 5985, 1140, 171, 18, 1;
475020, 98280, 17550, 2600, 300, 24, 1;
10295472, 1947792, 324632, 46376, 5456, 496, 31, 1;
260932815, 45379620, 7059052, 962598, 111930, 10660, 780, 39, 1; ...

Cf. A121441 (matrix inverse); A121412; variants: A122178, A121334, A121335; A122177 (dual).

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

Remarkably, row n of the matrix inverse (A121441) equals row n of A121412^(-n*(n+1)/2-3). 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'.

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'.

cp's OEIS Frontend

A121440 Matrix inverse of triangle A121335, where A121335(n,k) = C( n*(n+1)/2 + n-k + 1, n-k) for n>=k>=0.

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A121334 Triangle, read by rows, where T(n,k) = C( n*(n+1)/2 + n-k, n-k), for n>=k>=0.

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A121336 Triangle, read by rows, where T(n,k) = C( n*(n+1)/2 + n-k + 2, 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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