A121440 -id:A121440 - cp's OEIS frontend

A121335 Triangle, read by rows, where T(n,k) = C( n*(n+1)/2 + n-k + 1, n-k), for n>=k>=0.

1, 3, 1, 15, 5, 1, 120, 36, 8, 1, 1365, 364, 78, 12, 1, 20349, 4845, 969, 153, 17, 1, 376740, 80730, 14950, 2300, 276, 23, 1, 8347680, 1623160, 278256, 40920, 4960, 465, 30, 1, 215553195, 38320568, 6096454, 850668, 101270, 9880, 741, 38, 1, 6358402050

Offset: 0

Paul D. Hanna, Aug 29 2006

A triangle having similar properties and complementary construction is the dual triangle A122176.

			Triangle begins:
1;
3, 1;
15, 5, 1;
120, 36, 8, 1;
1365, 364, 78, 12, 1;
20349, 4845, 969, 153, 17, 1;
376740, 80730, 14950, 2300, 276, 23, 1;
8347680, 1623160, 278256, 40920, 4960, 465, 30, 1;
215553195, 38320568, 6096454, 850668, 101270, 9880, 741, 38, 1; ...

Cf. A121440 (matrix inverse); A121412; variants: A122178, A121334, A121336; A122176 (dual).

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

Remarkably, row n of the matrix inverse (A121440) equals row n of A121412^(-n*(n+1)/2-2). 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'.

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).

A121439 Matrix inverse of triangle A121334, where A121334(n,k) = C( n*(n+1)/2 + n-k, n-k) for n>=k>=0.

Original entry on oeis.org

1, -2, 1, -2, -4, 1, -14, 0, -7, 1, -143, -22, 11, -11, 1, -1928, -260, -40, 40, -16, 1, -32219, -3894, -385, -121, 99, -22, 1, -640784, -70644, -6496, -406, -406, 203, -29, 1, -14753528, -1502940, -128723, -9583, 259, -1184, 370, -37, 1, -385500056, -36631962, -2947266, -205620, -14076, 3657, -2967, 621

Offset: 0

Paul D. Hanna, Aug 29 2006

A triangle having similar properties and complementary construction is the dual triangle A121435.

			Triangle, A121334^-1, begins:
1;
-2, 1;
-2, -4, 1;
-14, 0, -7, 1;
-143, -22, 11, -11, 1;
-1928, -260, -40, 40, -16, 1;
-32219, -3894, -385, -121, 99, -22, 1;
-640784, -70644, -6496, -406, -406, 203, -29, 1;
-14753528, -1502940, -128723, -9583, 259, -1184, 370, -37, 1; ...
Triangle A121412 begins:
1;
1, 1;
3, 1, 1;
18, 4, 1, 1;
170, 30, 5, 1, 1; ...
Row 3 of A121334^-1 equals row 3 of A121412^(-7), which begins:
1;
-7, 1;
7, -7, 1;
-14, 0, -7, 1; ...
Row 4 of A121334^-1 equals row 4 of A121412^(-11), which begins:
1;
-11, 1;
33, -11, 1;
-22, 22, -11, 1;
-143, -22, 11, -11, 1;...

Cf. A121334 (matrix inverse); A121412; variants: A121438, A121440, A121441; A121435 (dual).

/* Matrix Inverse of A121334 */ {T(n,k)=local(M=matrix(n+1,n+1,r,c,if(r>=c,binomial(r*(r-1)/2+r-c,r-c)))); return((M^-1)[n+1,k+1])}

T(n,k) = [A121412^(-n*(n+1)/2 - 1)](n,k) for n>=k>=0; i.e., row n of A121334^-1 equals row n of matrix power A121412^(-n*(n+1)/2 - 1).

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

A121335 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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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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A121439 Matrix inverse of triangle A121334, where A121334(n,k) = C( n*(n+1)/2 + n-k, 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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