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%I #7 Mar 12 2015 21:10:33
%S 0,1,-5,83,16110,-40097784,-388036363380,82804198261002036,
%T 50475967918183333160880,-711988160501968313699728393632,
%U -26438313284970847487368499812182785280,22571673265500745067336177578868612107537514880
%N Column 0 of the matrix logarithm (A111833) of triangle A111830, which shifts columns left and up under matrix 7th power; these terms are the result of multiplying the element in row n by n!.
%C Let q=7; the g.f. of column k of A111830^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(7^j*x)/(j+1).
%e A(x) = x - 5/2!*x^2 + 83/3!*x^3 + 16110/4!*x^4 - 40097784/5!*x^5 +...
%e where e.g.f. A(x) satisfies:
%e x/(1-x) = A(x) + A(x)*A(7*x)/2! + A(x)*A(7*x)*A(7^2*x)/3! +
%e A(x)*A(7*x)*A(7^2*x)*A(7^3*x)/4! + ...
%e Let G(x) be the g.f. of A111831 (column 1 of A111830), then
%e G(x) = 1 + 7*A(x) + 7^2*A(x)*A(7*x)/2! +
%e 7^3*A(x)*A(7*x)*A(7^2*x)/3! +
%e 7^4*A(x)*A(7*x)*A(7^2*x)*A(7^3*x)/4! + ...
%o (PARI) {a(n,q=7)=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. A111830 (triangle), A111831, A111833 (matrix log); A110505 (q=-1), A111814 (q=2), A111816 (q=3), A111819 (q=4), A111824 (q=5), A111829 (q=6), A111839 (q=8).
%K sign
%O 0,3
%A _Gottfried Helms_ and _Paul D. Hanna_, Aug 22 2005
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