A111976 Column 0 of triangle A111975, which shifts columns left and up under matrix square.

1, 1, 1, 4, 16, 96, 896, 13568, 345088, 15112192, 1159913472, 158164664320, 38737429987328, 17197276791701504, 13946909814794223616, 20801835304287183306752, 57394078732651064041930752

Offset: 0

Author

Paul D. Hanna, Aug 25 2005

Keywords

• nonn

Examples

			G.f. A(x) = 1 + x + x^2 + 4*x^3 + 16*x^4 + 96*x^5 + 896*x^6 +...
= 1 + L(x) + L(x)*L(2*x)/2! + L(x)*L(2*x)*L(2^2*x)/3! +...
where L(x) = x + 16/3!*x^3 + 1536/5!*x^5 - 319488/7!*x^7 +-...
		

Crossrefs

Cf. A111975 (triangle), A111979.

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);return(A[n+1,1]))}

Formula

G.f.: A(x) = 1 + Sum_{n>=1} (1/n!)*Product_{j=0..n-1} L(2^j*x) where L(x) satisfies: x-x^2 = Sum_{j>=1}(1-2^j*x)*Prod_{i=0..j-1}L(2^i*x); and L(x) equals the g.f. of column 0 of the matrix log of A111975 (A111979).