cp's OEIS Frontend

A111816 Column 0 of the matrix logarithm (A111815) of triangle A078122, which shifts columns left and up under matrix cube; these terms are the result of multiplying the element in row n by n!.

%I A111816 #5 Mar 30 2012 18:36:50
%S A111816 0,1,-1,-3,150,1236,-2555748,-64342116,5885700899760,442646611978752,
%T A111816 -1737387344860364226240,-367706581563500487774720,
%U A111816 60788555325888838346137808787840,34626906551623392401873575206240000,-237458311254822429335982538087618909465992960
%N A111816 Column 0 of the matrix logarithm (A111815) of triangle A078122, which shifts columns left and up under matrix cube; these terms are the result of multiplying the element in row n by n!.
%C A111816 Let q=3; the g.f. of column k of A078122^m (matrix power m) is: 1 + Sum_{n>=1} (m*q^k)^n/n! * Product_{j=0..n-1} A(q^j*x).
%F A111816 E.g.f. satisfies: x/(1-x) = Sum_{n>=1} Prod_{j=0..n-1} A(3^j*x)/(j+1).
%e A111816 E.g.f.: A(x) = x - 1/2!*x^2 - 3/3!*x^3 + 150/4!*x^4 + 1236/5!*x^5 +...
%e A111816 where e.g.f. A(x) satisfies:
%e A111816 x/(1-x) = A(x) + A(x)*A(3*x)/2! + A(x)*A(3*x)*A(3^2*x)/3! +
%e A111816 A(x)*A(3*x)*A(3^2*x)*A(3^3*x)/4! + ...
%e A111816 Let G(x) be the g.f. of A078124 (column 1 of A078122), then
%e A111816 G(x) = 1 + 3*A(x) + 3^2*A(x)*A(3*x)/2! +
%e A111816 3^3*A(x)*A(3*x)*A(3^2*x)/3! +
%e A111816 3^4*A(x)*A(3*x)*A(3^2*x)*A(3^3*x)/4! + ...
%o A111816 (PARI) {a(n,q=3)=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 A111816 Cf. A078122 (triangle), A078124, A111815 (matrix log); A110505 (q=-1), A111814 (q=2), A111819 (q=4), A111824 (q=5), A111829 (q=6), A111834 (q=7), A111839 (q=8).
%K A111816 sign
%O A111816 0,4
%A A111816 _Gottfried Helms_ and _Paul D. Hanna_, Aug 22 2005