A111825 Triangle P, read by rows, that satisfies [P^6](n,k) = P(n+1,k+1) for n>=k>=0, also [P^(6*m)](n,k) = [P^m](n+1,k+1) for all m, where [P^m](n,k) denotes the element at row n, column k, of the matrix power m of P, with P(0,k)=1 and P(k,k)=1 for all k>=0.

1, 1, 1, 1, 6, 1, 1, 96, 36, 1, 1, 6306, 3816, 216, 1, 1, 1883076, 1625436, 139536, 1296, 1, 1, 2700393702, 3121837776, 360839016, 5036256, 7776, 1, 1, 19324893252552, 28794284803908, 4200503990976, 78293629296, 181382976, 46656, 1

Offset: 0

Author

Gottfried Helms and Paul D. Hanna, Aug 22 2005

Keywords

• nonn
• tabl

Comments

Also P(n,k) = the partitions of (6^n - 6^(n-k)) into powers of 6 <= 6^(n-k).

Examples

			Let q=6; the g.f. of column k of matrix power P^m is:
1 + (m*q^k)*L(x) + (m*q^k)^2/2!*L(x)*L(q*x) +
(m*q^k)^3/3!*L(x)*L(q*x)*L(q^2*x) +
(m*q^k)^4/4!*L(x)*L(q*x)*L(q^2*x)*L(q^3*x) + ...
where L(x) satisfies:
x/(1-x) = L(x) + L(x)*L(q*x)/2! + L(x)*L(q*x)*L(q^2*x)/3! + ...
and L(x) = x - 4/2!*x^2 + 42/3!*x^3 + 7296/4!*x^4 +... (A111829).
Thus the g.f. of column 0 of matrix power P^m is:
1 + m*L(x) + m^2/2!*L(x)*L(6*x) + m^3/3!*L(x)*L(6*x)*L(6^2*x) +
m^4/4!*L(x)*L(6*x)*L(6^2*x)*L(6^3*x) + ...
Triangle P begins:
1;
1,1;
1,6,1;
1,96,36,1;
1,6306,3816,216,1;
1,1883076,1625436,139536,1296,1;
1,2700393702,3121837776,360839016,5036256,7776,1; ...
where P^6 shifts columns left and up one place:
1;
6,1;
96,36,1;
6306,3816,216,1; ...
		

Crossrefs

Cf. A111826 (column 1), A111827 (row sums), A111828 (matrix log); triangles: A110503 (q=-1), A078121 (q=2), A078122 (q=3), A078536 (q=4), A111820 (q=5), A111830 (q=7), A111835 (q=8).

Programs

PARI

P(n,k,q=6)=local(A=Mat(1),B);if(n
				

Formula

Let q=6; the g.f. of column k of P^m (ignoring leading zeros) equals: 1 + Sum_{n>=1} (m*q^k)^n/n! * Product_{j=0..n-1} L(q^j*x) where L(x) satisfies: x/(1-x) = Sum_{n>=1} Product_{j=0..n-1} L(q^j*x)/(j+1) and L(x) equals the g.f. of column 0 of the matrix log of P (A111829).