A111849 Column 0 of the matrix logarithm (A111848) of triangle A111845, which shifts columns left and up under matrix 4th power; these terms are the result of multiplying the element in row n by n!.
0, 1, 4, 56, 1728, -45696, -159401472, 387212983296, 14722642769657856, -783395638188945997824, -571756408840959817330851840, 603349161280921866200339538247680, 8390141848229920894318007084122311229440
Offset: 0
Keywords
Examples
E.g.f. A(x) = x + 4/2!*x^2 + 56/3!*x^3 + 1728/4!*x^4 - 45696/5!*x^5 - 159401472/6!*x^6 +... where A(x) satisfies: x = A(x) - A(x)*A(4*x)/2! + A(x)*A(4*x)*A(4^2*x)/3! - A(x)*A(4*x)*A(4^2*x)*A(4^3*x)/4! + ... also: Let G(x) be the g.f. of A111846 (column 0 of A111845), then G(x) = 1 + x + 4*x^2 + 40*x^3 + 1040*x^4 + 78240*x^5 +... = 1 + A(x) + A(x)*A(4*x)/2! + A(x)*A(4*x)*A(4^2*x)/3! + A(x)*A(4*x)*A(4^2*x)*A(4^3*x)/4! +...
Crossrefs
Programs
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PARI
{a(n,q=4)=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]))}
Formula
E.g.f. satisfies: x = Sum_{n>=1} -(-1)^n/n!*Prod_{j=0..n-1} A(4^j*x).
Comments