A111824 Column 0 of the matrix logarithm (A111823) of triangle A111820, which shifts columns left and up under matrix 5th power; these terms are the result of multiplying the element in row n by n!.

0, 1, -3, 16, 2814, -1092180, -3603928080, 58978973128440, 5974833380453777520, -3294186866481455009752320, -10279982482873484428390722523200, 175129088125361734252730927280177244800

Offset: 0

Author

Gottfried Helms and Paul D. Hanna, Aug 22 2005

Keywords

• sign

Comments

Let q=5; the g.f. of column k of A111820^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 - 3/2!*x^2 + 16/3!*x^3 + 2814/4!*x^4 - 1092180/5!*x^5 +...
where e.g.f. A(x) satisfies:
x/(1-x) = A(x) + A(x)*A(5*x)/2! + A(x)*A(5*x)*A(5^2*x)/3! +
A(x)*A(5*x)*A(5^2*x)*A(5^3*x)/4! + ...
Let G(x) be the g.f. of A111821 (column 1 of A111820), then
G(x) = 1 + 5*A(x) + 5^2*A(x)*A(5*x)/2! +
5^3*A(x)*A(5*x)*A(5^2*x)/3! +
5^4*A(x)*A(5*x)*A(5^2*x)*A(5^3*x)/4! + ...
		

Crossrefs

Cf. A111820 (triangle), A111821, A111823 (matrix log); A110505 (q=-1), A111814 (q=2), A111816 (q=3), A111819 (q=4), A111829 (q=6), A111834 (q=7), A111839 (q=8).

Programs

PARI

{a(n,q=5)=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(5^j*x)/(j+1).