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

Author

Paul D. Hanna, Aug 29 2006

Keywords

• sign
• tabl

Comments

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

Examples

			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; ...
		

Crossrefs

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

Programs

PARI

/* 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])}

Formula

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).