login
A135902
Triangle T, read by rows, where column k of T = column 0 of T^(k+1) for k>0, with column 0 of T = column 0 of T^3 shift right.
4
1, 1, 1, 3, 2, 1, 15, 8, 3, 1, 102, 47, 15, 4, 1, 860, 356, 102, 24, 5, 1, 8548, 3252, 860, 186, 35, 6, 1, 97094, 34448, 8548, 1736, 305, 48, 7, 1, 1234324, 412546, 97094, 18754, 3130, 465, 63, 8, 1, 17302880, 5488222, 1234324, 228658, 36630, 5212, 672, 80, 9, 1
OFFSET
0,4
COMMENTS
This is a variant of permutation triangle P = A094587, A094587(n,k) = n!/k!, which may be defined by: triangular matrix P where column k of P = column 0 of P^(k+1), with column 0 of P = column 0 of P^2 shift right.
FORMULA
Column k of T^(j+1) = column j of T^(k+1) for j>=0, k>=0.
EXAMPLE
Triangle T begins:
1;
1, 1;
3, 2, 1;
15, 8, 3, 1;
102, 47, 15, 4, 1;
860, 356, 102, 24, 5, 1;
8548, 3252, 860, 186, 35, 6, 1;
97094, 34448, 8548, 1736, 305, 48, 7, 1;
1234324, 412546, 97094, 18754, 3130, 465, 63, 8, 1;
17302880, 5488222, 1234324, 228658, 36630, 5212, 672, 80, 9, 1; ...
where column k of T = column 0 of T^(k+1)
with column 0 of T = column 0 of T^3 shift right.
Matrix square of T, T^2, begins:
1;
2, 1;
8, 4, 1;
47, 22, 6, 1;
356, 156, 42, 8, 1;
3252, 1343, 351, 68, 10, 1;
34448, 13493, 3415, 656, 100, 12, 1; ...
where column 0 of T^2 = column 1 of T.
Matrix cube of T, T^3, begins:
1;
3, 1;
15, 6, 1;
102, 42, 9, 1;
860, 351, 81, 12, 1;
8548, 3415, 807, 132, 15, 1;
97094, 37795, 8967, 1530, 195, 18, 1; ...
where column 0 of T^3 = column 2 of T = column 0 of T shift left;
also, column 1 of T^3 = column 2 of T^2.
PROG
(PARI) T(n, k)=if(k>n || n<0 || k<0, 0, if(k==n, 1, if(k==0, T(n+1, 2), sum(j=0, n-k, T(n-k, j)*T(j+k-1, k-1))); ); )
CROSSREFS
Cf. columns: A135903, A135904, A135905; variants: A091351, A094587.
Sequence in context: A111548 A140709 A109282 * A135876 A136217 A166884
KEYWORD
nonn,tabl
AUTHOR
Paul D. Hanna, Dec 18 2007
STATUS
approved