cp's OEIS Frontend

A111844 Column 0 of the matrix logarithm (A111843) of triangle A111840, which shifts columns left and up under matrix cube; these terms are the result of multiplying the element in row n by n!.

%I A111844 #4 Mar 30 2012 18:36:50
%S A111844 0,1,3,27,486,7776,-2423196,-97338996,5883879500784,548540050402080,
%T A111844 -1737375315124971951360,-405928706169160555680960,
%U A111844 60788545124934395018363657569920,36207408592259278909089966337224960,-237458310218887960183820317532070376189904640
%N A111844 Column 0 of the matrix logarithm (A111843) of triangle A111840, which shifts columns left and up under matrix cube; these terms are the result of multiplying the element in row n by n!.
%C A111844 Let q=3; the g.f. of column k of A111840^m (matrix power m) is: 1 + Sum_{n>=1} (m*q^k)^n/n! * Product_{j=0..n-1} A(q^j*x).
%F A111844 E.g.f. satisfies: x = -Sum_{n>=1} Prod_{j=0..n-1} -A(3^j*x)/(j+1).
%e A111844 E.g.f. A(x) = x + 3/2!*x^2 + 27/3!*x^3 + 486/4!*x^4 + 7776/5!*x^5
%e A111844 - 2423196/6!*x^6 - 97338996/7!*x^7 +...
%e A111844 where A(x) satisfies:
%e A111844 x = A(x) - A(x)*A(3*x)/2! + A(x)*A(3*x)*A(3^2*x)/3!
%e A111844 - A(x)*A(3*x)*A(3^2*x)*A(3^3*x)/4! + ...
%e A111844 also:
%e A111844 Let G(x) be the g.f. of A111841 (column 0 of A111840), then
%e A111844 G(x) = 1 + x + 3*x^2 + 18*x^3 + 216*x^4 + 5589*x^5 + 336555*x^6 +...
%e A111844 = 1 + A(x) + A(x)*A(3*x)/2! + A(x)*A(3*x)*A(3^2*x)/3!
%e A111844 + A(x)*A(3*x)*A(3^2*x)*A(3^3*x)/4! +...
%o A111844 (PARI) {a(n,q=3)=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 A111844 Cf. A111843 (matrix log), A111840 (triangle), A111841, A111816 (variant), A111941 (q=-1), A111843 (q=3), A111848 (q=4).
%K A111844 sign
%O A111844 0,3
%A A111844 _Paul D. Hanna_, Aug 23 2005