A111829 Column 0 of the matrix logarithm (A111828) of triangle A111825, which shifts columns left and up under matrix 6th power; these terms are the result of multiplying the element in row n by n!.

0, 1, -4, 42, 7296, -7931976, -45557382240, 3064554175021200, 801993619807364206080, -2618439032548254776387771520, -30580166025709706974876961026475520, 4440597519115996836838709580481861376121600

Offset: 0

Author

Gottfried Helms and Paul D. Hanna, Aug 22 2005

Keywords

• sign

Comments

Let q=6; the g.f. of column k of A111825^m (matrix power m) is: 1 + Sum_{n>=1} (m*q^k)^n/n! * Product_{j=0..n-1} A(q^j*x).

Examples

			A(x) = x - 4/2!*x^2 + 42/3!*x^3 + 7296/4!*x^4 - 7931976/5!*x^5 +...
where e.g.f. A(x) satisfies:
x/(1-x) = A(x) + A(x)*A(6*x)/2! + A(x)*A(6*x)*A(6^2*x)/3! +
A(x)*A(6*x)*A(6^2*x)*A(6^3*x)/4! + ...
Let G(x) be the g.f. of A111826 (column 1 of A111825), then
G(x) = 1 + 6*A(x) + 6^2*A(x)*A(6*x)/2! +
6^3*A(x)*A(6*x)*A(6^2*x)/3! +
6^4*A(x)*A(6*x)*A(6^2*x)*A(6^3*x)/4! + ...
		

Crossrefs

Cf. A111825 (triangle), A111826, A111828 (matrix log); A110505 (q=-1), A111814 (q=2), A111816 (q=3), A111819 (q=4), A111824 (q=5), A111834 (q=7), A111839 (q=8).

Programs

PARI

{a(n,q=6)=local(A=x/(1-x+x*O(x^n)));for(i=1,n, A=x/(1-x)/(1+sum(j=1,n,prod(k=1,j,subst(A,x,q^k*x))/(j+1)!))); return(n!*polcoeff(A,n))}

Formula

E.g.f. satisfies: x/(1-x) = Sum_{n>=1} Prod_{j=0..n-1} A(6^j*x)/(j+1).