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%I #7 Mar 12 2015 19:50:32
%S 0,1,-6,142,31800,-159468264,-2481298801008,1414130111428687344,
%T 1827317023092830201950080,-89946874545119714361987192509568,
%U -9262235489215916508714844705185660161280
%N Column 0 of the matrix logarithm (A111838) of triangle A111835, which shifts columns left and up under matrix 8th power; these terms are the result of multiplying the element in row n by n!.
%C Let q=8; 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).
%F E.g.f. satisfies: x/(1-x) = Sum_{n>=1} Prod_{j=0..n-1} A(8^j*x)/(j+1).
%e A(x) = x - 6/2!*x^2 + 142/3!*x^3 + 31800/4!*x^4 - 159468264/5!*x^5 +...
%e where e.g.f. A(x) satisfies:
%e x/(1-x) = A(x) + A(x)*A(8*x)/2! + A(x)*A(8*x)*A(8^2*x)/3! +
%e A(x)*A(8*x)*A(8^2*x)*A(8^3*x)/4! + ...
%e Let G(x) be the g.f. of A111836 (column 1 of A111835), then
%e G(x) = 1 + 8*A(x) + 8^2*A(x)*A(8*x)/2! +
%e 8^3*A(x)*A(8*x)*A(8^2*x)/3! +
%e 8^4*A(x)*A(8*x)*A(8^2*x)*A(8^3*x)/4! + ...
%o (PARI) {a(n,q=8)=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))}
%Y Cf. A111835 (triangle), A111836, A111838 (matrix log); A110505 (q=-1), A111814 (q=2), A111816 (q=3), A111819 (q=4), A111824 (q=5), A111829 (q=6), A111834 (q=7).
%K sign
%O 0,3
%A _Gottfried Helms_ and _Paul D. Hanna_, Aug 22 2005
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