cp's OEIS Frontend

A111819 Column 0 of the matrix logarithm (A111818) of triangle A078536, which shifts columns left and up under matrix 4th power; these terms are the result of multiplying the element in row n by n!.

%I A111819 #7 Mar 14 2015 10:02:51
%S A111819 0,1,-2,2,840,-76056,-158761104,390564896784,14713376473366656,
%T A111819 -783793232940393380736,-571732910947761663424746240,
%U A111819 603368029500890443054004423520000,8390120127886533420753746115877557580800
%N A111819 Column 0 of the matrix logarithm (A111818) of triangle A078536, 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 A111819 Let q=4; the g.f. of column k of A078536^m (matrix power m) is: 1 + Sum_{n>=1} (m*q^k)^n/n! * Product_{j=0..n-1} A(q^j*x).
%F A111819 E.g.f. satisfies: x/(1-x) = Sum_{n>=1} Prod_{j=0..n-1} A(4^j*x)/(j+1).
%e A111819 A(x) = x - 2/2!*x^2 + 2/3!*x^3 + 840/4!*x^4 - 76056/5!*x^5 +...
%e A111819 where e.g.f. A(x) satisfies:
%e A111819 x/(1-x) = A(x) + A(x)*A(4*x)/2! + A(x)*A(4*x)*A(4^2*x)/3! +
%e A111819 A(x)*A(4*x)*A(4^2*x)*A(4^3*x)/4! + ...
%e A111819 Let G(x) be the g.f. of A111817 (column 1 of A078536), then
%e A111819 G(x) = 1 + 4*A(x) + 4^2*A(x)*A(4*x)/2! +
%e A111819 4^3*A(x)*A(4*x)*A(4^2*x)/3! +
%e A111819 4^4*A(x)*A(4*x)*A(4^2*x)*A(4^3*x)/4! + ...
%o A111819 (PARI) {a(n,q=4)=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 A111819 Cf. A078536 (triangle), A111817, A111818 (matrix log); A110505 (q=-1), A111814 (q=2), A111816 (q=3), A111824 (q=5), A111829 (q=6), A111834 (q=7), A111839 (q=8).
%K A111819 sign
%O A111819 0,3
%A A111819 _Gottfried Helms_ and _Paul D. Hanna_, Aug 22 2005