OFFSET
0,3
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.
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).
EXAMPLE
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 +...
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 *)
PROG
(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), ", "))
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Apr 05 2012
STATUS
approved