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%I #6 Mar 14 2015 10:04:02
%S 0,1,4,56,1728,-45696,-159401472,387212983296,14722642769657856,
%T -783395638188945997824,-571756408840959817330851840,
%U 603349161280921866200339538247680,8390141848229920894318007084122311229440
%N Column 0 of the matrix logarithm (A111848) of triangle A111845, which shifts columns left and up under matrix 4th power; these terms are the result of multiplying the element in row n by n!.
%C Let q=4; the g.f. of column k of A111845^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 = Sum_{n>=1} -(-1)^n/n!*Prod_{j=0..n-1} A(4^j*x).
%e E.g.f. A(x) = x + 4/2!*x^2 + 56/3!*x^3 + 1728/4!*x^4
%e - 45696/5!*x^5 - 159401472/6!*x^6 +...
%e where A(x) satisfies:
%e x = A(x) - A(x)*A(4*x)/2! + A(x)*A(4*x)*A(4^2*x)/3!
%e - A(x)*A(4*x)*A(4^2*x)*A(4^3*x)/4! + ...
%e also:
%e Let G(x) be the g.f. of A111846 (column 0 of A111845), then
%e G(x) = 1 + x + 4*x^2 + 40*x^3 + 1040*x^4 + 78240*x^5 +...
%e = 1 + A(x) + A(x)*A(4*x)/2! + A(x)*A(4*x)*A(4^2*x)/3!
%e + A(x)*A(4*x)*A(4^2*x)*A(4^3*x)/4! +...
%o (PARI) {a(n,q=4)=local(A=Mat(1),B);if(n<0,0, for(m=1,n+1,B=matrix(m,m);for(i=1,m, for(j=1,i, if(j==i,B[i,j]=1,if(j==1,B[i,j]=(A^q)[i-1,1], B[i,j]=(A^q)[i-1,j-1]));));A=B); B=sum(i=1,#A,-(A^0-A)^i/i);return(n!*B[n+1,1]))}
%Y Cf. A111848 (matrix log), A111845 (triangle), A111846, A111821 (variant), A111942 (q=-1), A111811 (q=2), A111844 (q=3).
%K sign
%O 0,3
%A _Paul D. Hanna_, Aug 23 2005
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