A111815 Matrix log of triangle A078122, which shifts columns left and up under matrix cube; these terms are the result of multiplying each element in row n and column k by (n-k)!.

0, 1, 0, -1, 3, 0, -3, -3, 9, 0, 150, -9, -9, 27, 0, 1236, 450, -27, -27, 81, 0, -2555748, 3708, 1350, -81, -81, 243, 0, -64342116, -7667244, 11124, 4050, -243, -243, 729, 0, 5885700899760, -193026348, -23001732, 33372, 12150, -729, -729, 2187, 0

Offset: 0

Author

Gottfried Helms and Paul D. Hanna, Aug 22 2005

Keywords

• frac
• sign
• tabl

Comments

Column k equals 3^k multiplied by column 0 (A111816) when ignoring zeros above the diagonal.

Examples

			Matrix log of A078122, with factorial denominators, begins:
0;
1/1!, 0;
-1/2!, 3/1!, 0;
-3/3!, -3/2!, 9/1!, 0;
150/4!, -9/3!, -9/2!, 27/1!, 0;
1236/5!, 450/4!, -27/3!, -27/2!, 81/1!, 0;
-2555748/6!, 3708/5!, 1350/4!, -81/3!, -81/2!, 243/1!, 0; ...
		

Crossrefs

Cf. A078122, A111816 (column 0), A111840 (variant); log matrices: A110504 (q=-1), A111813 (q=2), A111818 (q=4), A111823 (q=5), A111828 (q=6), A111833 (q=7), A111838 (q=8).

Programs

PARI

T(n,k,q=3)=local(A=Mat(1),B);if(n
				

Formula

T(n, k) = 3^k*T(n-k, 0) = A111816(n-k) for n>=k>=0.