cp's OEIS Frontend

A111811 Column 0 of the matrix logarithm (A111810) of triangle A098539, which shifts columns left and up under matrix square; these terms are the result of multiplying the element in row n by n!.

%I A111811 #4 Mar 30 2012 18:36:50
%S A111811 0,1,2,10,88,1096,11856,-402480,-1891968,36024603264,359905478400,
%T A111811 -53686393014816000,-644141701131494400,1790653231402788752593920,
%U A111811 25068910772059830672353280,-1280832036591718248285105113241600
%N A111811 Column 0 of the matrix logarithm (A111810) of triangle A098539, which shifts columns left and up under matrix square; these terms are the result of multiplying the element in row n by n!.
%C A111811 Let q=2; the g.f. of column k of A098539^m (matrix power m) is: 1 + Sum_{n>=1} (m*q^k)^n/n! * Product_{j=0..n-1} A(q^j*x).
%F A111811 E.g.f. satisfies: x = -Sum_{n>=1} Prod_{j=0..n-1} -A(2^j*x)/(j+1), also: x/(1+x) = Sum_{n>=1} (-2)^(n-1)*Prod_{j=0..n-1} A(2^j*x)/(j+1).
%e A111811 A(x) = x + 2/2!*x^2 + 10/3!*x^3 + 88/4!*x^4 + 1096/5!*x^5 +...
%e A111811 where e.g.f. A(x) satisfies:
%e A111811 x = A(x) - A(x)*A(2*x)/2! + A(x)*A(2*x)*A(2^2*x)/3! - A(x)*A(2*x)*A(2^2*x)*A(2^3*x)/4! + ...
%e A111811 also:
%e A111811 x/(1+x) = A(x) - 2*A(x)*A(2*x)/2! + 2^2*A(x)*A(2*x)*A(2^2*x)/3! - 2^3*A(x)*A(2*x)*A(2^2*x)*A(2^3*x)/4! +...
%e A111811 Let G(x) be the g.f. of A002449 (column 1 of A098539), then
%e A111811 (G(x)-1)/x = 1 + 2*x + 6*x^2 + 26*x^3 + 166*x^4 + 1626*x^5 +...
%e A111811 = 1 + 2*A(x) + 2^2*A(x)*A(2*x)/2! + 2^3*A(x)*A(2*x)*A(2^2*x)/3! + 2^4*A(x)*A(2*x)*A(2^2*x)*A(2^3*x)/4! +...
%o A111811 (PARI) {a(n,q=2)=local(A=x+x*O(x^n));for(i=1,n, A=x/(1+sum(j=1,n,prod(k=1,j,-subst(A,x,q^k*x))/(j+1)!))); return(n!*polcoeff(A,n))}
%Y A111811 Cf. A111810 (matrix log), A098539 (triangle), A002449, A111814 (variant), A111941 (q=-1), A111843 (q=3), A111848 (q=4).
%K A111811 sign
%O A111811 0,3
%A A111811 _Paul D. Hanna_, Aug 22 2005