A132620 Triangle T, read by rows, where antidiagonal k of T = antidiagonal k-1 of T^k (after appending '1' for even k) for k>0, with T(n,n)=1 for n>=0.

1, 1, 1, 2, 1, 1, 9, 4, 1, 1, 76, 30, 6, 1, 1, 1015, 350, 63, 8, 1, 1, 18996, 5740, 952, 108, 10, 1, 1, 461160, 123536, 19026, 2010, 165, 12, 1, 1, 13810056, 3321624, 477612, 47685, 3652, 234, 14, 1, 1, 492785919, 107639880, 14523179, 1379928, 100529, 6006

Offset

0,4

Links

Table of n, a(n) for n=0..50.

Example

Triangle begins:
1;
1, 1;
2, 1, 1;
9, 4, 1, 1;
76, 30, 6, 1, 1;
1015, 350, 63, 8, 1, 1;
18996, 5740, 952, 108, 10, 1, 1;
461160, 123536, 19026, 2010, 165, 12, 1, 1;
13810056, 3321624, 477612, 47685, 3652, 234, 14, 1, 1;
492785919, 107639880, 14523179, 1379928, 100529, 6006, 315, 16, 1, 1; ...
GENERATE T FROM MATRIX POWERS OF T.
Matrix cube, T^3, begins:
1;
3, (1);
(9), 3, 1;
46, 15, 3, 1; ...
where diagonal 3 of T = [9,1] = diagonal 2 of T^3.
Matrix fourth power, T^4, begins:
1;
4, 1;
14, (4), 1;
(76), 22, 4, 1; ...
where diagonal 4 of T = [76,4,1] = diagonal 3 of T^4 append '1'.
Matrix fifth power, T^5, begins:
1,
5, 1,
20, 5, (1),
115, (30), 5, 1,
(1015), 260, 40, 5, 1, ...
where diagonal 5 of T = [1015,30,1] = diagonal 4 of T^5.
Matrix sixth power, T^6, begins:
1;
6, 1;
27, 6, 1;
164, 39, (6), 1;
1476, (350), 51, 6, 1;
(18996), 4350, 608, 63, 6, 1; ...
where diagonal 6 of T = [18996,350,6,1] = diagonal 5 of T^6 append '1'.

Prog

(PARI) T(n, k)=local(M=matrix(n+1, n+1)); for(r=1, n+1, for(c=1, r, M[r, c]=if(r==c || r==c+1, 1, (M^(r+c-2))[r-1, c]))); M[n+1, k+1]

Crossrefs

Cf. columns: A132621, A132622.

Sequence in context: A167015 A124522 A016540 * A156883 A368924 A019803

Adjacent sequences: A132617 A132618 A132619 * A132621 A132622 A132623

Keyword

nonn,tabl

Author

Paul D. Hanna, Aug 24 2007

Status

approved