A111978 Matrix log of triangle A111975, which shifts columns left and up under matrix square; these terms are the result of multiplying each element in row n and column k by (n-k)!.

0, 1, 0, 0, 2, 0, 16, 0, 4, 0, 0, 32, 0, 8, 0, 1536, 0, 64, 0, 16, 0, 0, 3072, 0, 128, 0, 32, 0, -319488, 0, 6144, 0, 256, 0, 64, 0, 0, -638976, 0, 12288, 0, 512, 0, 128, 0, 36007575552, 0, -1277952, 0, 24576, 0, 1024, 0, 256, 0, 0, 72015151104, 0, -2555904, 0, 49152, 0, 2048, 0, 512, 0

Offset

0,5

Comments

Column k equals 2^k multiplied by column 0 (A111979) when ignoring zeros above the diagonal.

Links

Table of n, a(n) for n=0..65.

Formula

T(n, k) = 2^k*T(n-k, 0) = 2^k*A111979(n-k) for n>=k>=0.

Example

Matrix log of A111975, with factorial denominators, begins:
0;
1/1!, 0;
0/2!, 2/1!, 0;
16/3!, 0/2!, 4/1!, 0;
0/4!, 32/3!, 0/2!, 8/1!, 0;
1536/5!, 0/4!, 64/3!, 0/2!, 16/1!, 0;
0/6!, 3072/5!, 0/4!, 128/3!, 0/2!, 32/1!, 0;
-319488/7!, 0/6!, 6144/5!, 0/4!, 256/3!, 0/2!, 64/1!, 0; ...

Prog

(PARI) T(n, k, q=2)=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, B[i, j]=1, if(j==1, B[i, j]=if(i>2, (A^q)[i-1, 2], 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. A111975 (triangle), A111979 (column 0).

Sequence in context: A232423 A074031 A086261 * A146558 A364514 A025600

Adjacent sequences: A111975 A111976 A111977 * A111979 A111980 A111981

Keyword

frac,sign,tabl

Author

Paul D. Hanna, Aug 25 2005

Status

approved