OFFSET
0,2
FORMULA
Let [Q^m]_k denote column k of matrix power Q^m,
so that triangular matrix Q may be defined by
[Q]_k = [P^(2*k+2)]_0, for k>=0, where
the dual triangular matrix P = A113340 is defined by
[P]_k = [P^(2*k+1)]_0, for k>=0.
Then, amazingly, powers of P and Q satisfy:
[P^(2*j+1)]_k = [P^(2*k+1)]_j,
[P^(2*j+2)]_k = [Q^(2*k+1)]_j,
[Q^(2*j+2)]_k = [Q^(2*k+2)]_j,
for all j>=0, k>=0.
Also, we have the column transformations:
P^2 * [P]_k = [P]_{k+1},
P^2 * [Q]_k = [Q]_{k+1},
Q^2 * [P^2]_k = [P^2]_{k+1},
Q^2 * [Q^2]_k = [Q^2]_{k+1},
for all k>=0.
EXAMPLE
Triangle Q begins:
1;
2,1;
5,4,1;
19,22,6,1;
113,166,51,8,1;
966,1671,561,92,10,1;
10958,21510,7726,1324,145,12,1;
156700,341463,129406,23010,2575,210,14,1;
2727794,6496923,2572892,471724,53935,4434,287,16,1;
56306696,144856710,59525136,11198006,1305070,108593,7021,376,18,1;
Matrix square Q^2 begins:
1;
4,1;
18,8,1;
112,68,12,1;
965,712,150,16,1;
10957,9270,2184,264,20,1; ...
where Q^2 transforms column k of Q^2 into column k+1:
at k=0, [Q^2]*[1,4,18,112,965,...] = [1,8,68,712,9270,...];
at k=1, [Q^2]*[1,8,68,712,9270,...] =
[1,12,150,2184,37523,...].
PROG
(PARI) Q(n, k)=local(A, B); A=matrix(1, 1); A[1, 1]=1; for(m=2, n+1, B=matrix(m, m); for(i=1, m, for(j=1, i, if(i<3 || j==i || j>m-1, B[i, j]=1, if(j==1, B[i, 1]=1, B[i, j]=(A^(2*j-1))[i-j+1, 1])); )); A=B); (A^(2*k+2))[n-k+1, 1]
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Paul D. Hanna, Nov 08 2005
STATUS
approved