A121438 -id:A121438 - cp's OEIS frontend

A122178 Triangle, read by rows, where T(n,k) = C( n*(n+1)/2 + n-k - 1, n-k), for n>=k>=0.

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'.

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).

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

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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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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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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