A111818 Matrix log of triangle A078536, which shifts columns left and up under matrix 4th power; these terms are the result of multiplying each element in row n and column k by (n-k)!.

0, 1, 0, -2, 4, 0, 2, -8, 16, 0, 840, 8, -32, 64, 0, -76056, 3360, 32, -128, 256, 0, -158761104, -304224, 13440, 128, -512, 1024, 0, 390564896784, -635044416, -1216896, 53760, 512, -2048, 4096, 0, 14713376473366656, 1562259587136, -2540177664, -4867584, 215040, 2048, -8192, 16384, 0

Offset

0,4

Comments

Column k equals 4^k multiplied by column 0 (A111819) when ignoring zeros above the diagonal.

Links

Table of n, a(n) for n=0..44.

Formula

T(n, k) = 4^k*T(n-k, 0) = A111819(n-k) for n>=k>=0.

Example

Matrix log of A078536, with factorial denominators, begins:
0;
1/1!, 0;
-2/2!, 4/1!, 0;
2/3!, -8/2!, 16/1!, 0;
840/4!, 8/3!, -32/2!, 64/1!, 0;
-76056/5!, 3360/4!, 32/3!, -128/2!, 256/1!, 0;
-158761104/6!, -304224/5!, 13440/4!, 128/3!, -512/2!, 1024/1!, 0;

Prog

(PARI) T(n, k, q=4)=local(A=Mat(1), B); if(n<k || k<0, 0, for(m=1, n+1, B=matrix(m, m); for(i=1, m, for(j=1, i, if(j==i || j==1, B[i, j]=1, B[i, j]=(A^q)[i-1, j-1]); )); A=B); B=sum(i=1, #A, -(A^0-A)^i/i); return((n-k)!*B[n+1, k+1]))

Crossrefs

Cf. A078536, A111819 (column 0), A111845 (variant); log matrices: A110504 (q=-1), A111813 (q=2), A111815 (q=3), A111823 (q=5), A111828 (q=6), A111833 (q=7), A111838 (q=8).

Sequence in context: A256014 A256280 A153182 * A007631 A281765 A155517

Adjacent sequences: A111815 A111816 A111817 * A111819 A111820 A111821

Keyword

frac,sign,tabl

Author

Gottfried Helms and Paul D. Hanna, Aug 22 2005

Status

approved