A111979 Column 0 of the matrix logarithm (A111978) of triangle A111975, which shifts columns left and up under matrix square; these terms are the result of multiplying the element in row n by n!.

0, 1, 0, 16, 0, 1536, 0, -319488, 0, 36007575552, 0, -53682434054553600, 0, 1790644857560674043166720, 0, -1280831660558056667387645027942400, 0, 18961467116136182692294341450867551502336000, 0

Offset: 0

Author

Paul D. Hanna, Aug 25 2005

Keywords

• sign

Comments

Let q=2; the g.f. of column k of A111975^m (matrix power m) is: 1 + Sum_{n>=1} (m*q^k)^n/n! * Product_{j=0..n-1} A(q^j*x).

Examples

			E.g.f. A(x) = x + 16/3!*x^3 + 1536/5!*x^5 - 319488/7!*x^7
+ 36007575552/9!*x^9 - 53682434054553600/11!*x^11 +...
where A(x) satisfies:
x*(1-x) = (1-2*x)*A(x) + (1-2^2*x)*A(x)*A(2*x)/2!
+ (1-2^3*x)*A(x)*A(2*x)*A(2^2*x)/3! +...
		

Crossrefs

Cf. A111978 (matrix log), A111975 (triangle), A111976, A111811 (variant), A111814 (variant).

Programs

PARI

{a(n,q=2)=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]=if(i>2,(A^q)[i-1,2],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]))}

Formula

E.g.f. A(x): x-x^2 = Sum_{j>=1}(1-2^j*x)/j!*Prod_{i=0..j-1}A(2^i*x). E.g.f. A(x): x+x^2 = Sum_{j>=1}(1-4^j*x^2)/j!*Prod_{i=0..j-1}A(2^i*x).