A121439 -id:A121439 - cp's OEIS frontend

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

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'.

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).

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

Original entry on oeis.org

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).

A121441 Matrix inverse of triangle A121336, where A121336(n,k) = C( n*(n+1)/2 + n-k + 2, n-k) for n>=k>=0.

Original entry on oeis.org

1, -4, 1, 3, -6, 1, -12, 9, -9, 1, -117, -26, 26, -13, 1, -1656, -216, -69, 63, -18, 1, -28506, -3396, -294, -212, 132, -24, 1, -578274, -63116, -5766, -124, -620, 248, -31, 1, -13504179, -1365546, -116116, -8892, 1170, -1612, 429, -39, 1, -356633784, -33696726, -2696316, -186860, -15000, 6228, -3736

Offset: 0

Paul D. Hanna, Aug 29 2006

A triangle having similar properties and complementary construction is the dual triangle A121437.

			Triangle, A121336^-1, begins:
1;
-4, 1;
3, -6, 1;
-12, 9, -9, 1;
-117, -26, 26, -13, 1;
-1656, -216, -69, 63, -18, 1;
-28506, -3396, -294, -212, 132, -24, 1;
-578274, -63116, -5766, -124, -620, 248, -31, 1;
-13504179, -1365546, -116116, -8892, 1170, -1612, 429, -39, 1; ...
Triangle A121412 begins:
1;
1, 1;
3, 1, 1;
18, 4, 1, 1;
170, 30, 5, 1, 1; ...
Row 3 of A121336^-1 equals row 3 of A121412^(-9), which begins:
1;
-9, 1;
18, -9, 1;
-12, 9, -9, 1; ...
Row 4 of A121336^-1 equals row 4 of A121412^(-13), which begins:
1;
-13, 1;
52, -13, 1;
-52, 39, -13, 1;
-117, -26, 26, -13, 1; ...

Cf. A121336 (matrix inverse); A121412; variants: A121438, A121439, A121440; A121437 (dual).

/* Matrix Inverse of A121336 */ {T(n,k)=local(M=matrix(n+1,n+1,r,c,if(r>=c,binomial(r*(r-1)/2+r-c+2,r-c)))); return((M^-1)[n+1,k+1])}

T(n,k) = [A121412^(-n*(n+1)/2 - 3)](n,k) for n>=k>=0; i.e., row n of A121336^-1 equals row n of matrix power A121412^(-n*(n+1)/2 - 3).

cp's OEIS Frontend

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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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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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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A121441 Matrix inverse of triangle A121336, where A121336(n,k) = C( n*(n+1)/2 + n-k + 2, n-k) for n>=k>=0.

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