A181998 G.f. satisfies: x = Sum_{n>=1} 1/A(x)^(4*n) * Product_{k=1..n} (1 - 1/A(x)^k).

1, 1, 3, 18, 124, 935, 7443, 61510, 522467, 4532452, 39985628, 357641094, 3235846003, 29565353095, 272429349163, 2528938553028, 23629834081955, 222080711420655, 2098112946860819, 19915641133236764, 189853287434733709, 1816924035668823659, 17450483777418686431

Offset: 0

Author

Paul D. Hanna, Apr 05 2012

Keywords

• nonn

Comments

Compare the g.f. to the identity:

G(x) = Sum_{n>=0} 1/G(x)^n * Product_{k=1..n} (1 - 1/G(x)^k)

which holds for all power series G(x) such that G(0)=1.

Examples

			G.f.: A(x) = 1 + x + 3*x^2 + 18*x^3 + 124*x^4 + 935*x^5 + 7443*x^6 +...
The g.f. satisfies:
x = (A(x)-1)/A(x)^5 + (A(x)-1)*(A(x)^2-1)/A(x)^11 + (A(x)-1)*(A(x)^2-1)*(A(x)^3-1)/A(x)^18 + (A(x)-1)*(A(x)^2-1)*(A(x)^3-1)*(A(x)^4-1)/A(x)^26 + (A(x)-1)*(A(x)^2-1)*(A(x)^3-1)*(A(x)^4-1)*(A(x)^5-1)/A(x)^35 +...
		

Crossrefs

Cf. A001002, A181997, A209441, A209442, A214694 (variant).

Programs

Mathematica

nmax = 20; aa = ConstantArray[0,nmax]; aa[[1]] = 1; Do[AGF = 1+Sum[aa[[n]]*x^n,{n,1,j-1}]+koef*x^j; sol=Solve[SeriesCoefficient[Sum[Product[(1-1/AGF^m)/AGF^4,{m,1,k}],{k,1,j}],{x,0,j}]==0,koef][[1]]; aa[[j]]=koef/.sol[[1]],{j,2,nmax}]; Flatten[{1,aa}] (* Vaclav Kotesovec, Dec 01 2014 *)
CoefficientList[1+InverseSeries[Series[x - 3*x^2 + 11*x^4 + x^5 - 30*x^6 - 42*x^7 - 26*x^8 - 8*x^9 - x^10, {x, 0, 20}], x],x] (* Vaclav Kotesovec, Dec 01 2014 *)

PARI

{a(n)=if(n<0,0,polcoeff(1 + serreverse(x - 3*x^2 + 11*x^4 + x^5 - 30*x^6 - 42*x^7 - 26*x^8 - 8*x^9 - x^10 +x^2*O(x^n)),n))}

PARI

{a(n)=local(A=[1,1]);for(i=1,n,A=concat(A,0);A[#A]=-polcoeff(sum(m=1,#A,1/Ser(A)^(4*m)*prod(k=1,m,1-1/Ser(A)^k)),#A-1));A[n+1]}
for(n=0,25,print1(a(n),", "))

Formula

G.f. satisfies: 1+x = A(y) where y = x - 3*x^2 + 11*x^4 + x^5 - 30*x^6 - 42*x^7 - 26*x^8 - 8*x^9 - x^10.

G.f. satisfies: x = Sum_{n>=1} 1/A(x)^(n*(n+9)/2) * Product_{k=1..n} (A(x)^k - 1).