A119459 Decomposition of function F = x/(1-x) into functions of the form [x + a(n)*x^n]: x = ...o x+a(n)*x^n o...o x+a(3)*x^3 o x+a(2)*x^2 o a(1)*x o F.

1, -1, 1, 2, 6, 10, 27, 54, 253, 312, 1116, 2194, 6157, 15552, 54514, 89014, 687252, 964080, 2577597, 6822210, 22066744, 38505658, 119635413, 289026266, 517893326, 2410443144, 9800259872, 22806899334, 87955801599, 189512770532, 416815519198, 1126216162158, 9805754835374

Offset: 1

Author

Paul D. Hanna, May 20 2006

Keywords

• sign

Examples

			Iterated decompositions of F=x/(1-x) into [x + a(n)*x^n]:
x = ... o x+6*x^5 o x+2*x^4 o x+1*x^3 o x-1*x^2 o 1*x o F.
These decompositions get closer to x at each iteration:
(1) 1*x o F = x/(1-x) = x + x^2 + x^3 + x^4 + x^5 +...
(2) x-1*x^2 o 1*x o F =
x - x^3 - 2*x^4 - 3*x^5 - 4*x^6 - 5*x^7 - 6*x^8 -...
(3) x+1*x^3 o x-1*x^2 o 1*x o F =
x - 2*x^4 - 6*x^5 - 10*x^6 - 11*x^7 - 6*x^8 + 7*x^9 +...
(4) x+2*x^4 o x+1*x^3 o x-1*x^2 o 1*x o F =
x - 6*x^5 - 10*x^6 - 27*x^7 - 54*x^8 - 73*x^9 +...
(5) x+6*x^5 o x+2*x^4 o x+1*x^3 o x-1*x^2 o 1*x o F =
x - 10*x^6 - 27*x^7 - 54*x^8 - 253*x^9 +...
		

Crossrefs

Cf. A119460 (composition of x/(1-x)).

Programs

PARI

{a(n)=local(F=x/(1-x+x*O(x^n)));if(n<1,0,if(n==1,1, for(k=2,n,c=-polcoeff(F,k);F=subst(x+c*x^k,x,F););return(c)))}