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%I #9 Jan 03 2018 15:54:17
%S 1,1,5,45,470,5365,64766,813012,10505163,138800397,1866712401,
%T 25470265992,351717013269,4906153922941,69030042202001,
%U 978531875343171,13961726654230994,200351151383453293,2889692388200640136,41867983817065377259,609091785100828769195
%N G.f. satisfies: x = Sum_{n>=1} 1/A(x)^(6*n) * Product_{k=1..n} (1 - 1/A(x)^k).
%C Compare the g.f. to the identity:
%C G(x) = Sum_{n>=0} 1/G(x)^n * Product_{k=1..n} (1 - 1/G(x)^k)
%C which holds for all power series G(x) such that G(0)=1.
%H G. C. Greubel, <a href="/A209442/b209442.txt">Table of n, a(n) for n = 0..835</a>
%F G.f. satisfies: 1+x = A(y) where y = x - 5*x^2 + 5*x^3 + 30*x^4 - 65*x^5 - 191*x^6 + 378*x^7 + 1557*x^8 + 103*x^9 - 8551*x^10 - 23911*x^11 - 37958*x^12 - 41831*x^13 - 34156*x^14 - 21179*x^15 - 10015*x^16 - 3571*x^17 - 933*x^18 - 169*x^19 - 19*x^20 - x^21.
%F G.f. satisfies: x = Sum_{n>=1} 1/A(x)^(n*(n+13)/2) * Product_{k=1..n} (A(x)^k - 1).
%e G.f.: A(x) = 1 + x + 5*x^2 + 45*x^3 + 470*x^4 + 5365*x^5 + 64766*x^6 +...
%e The g.f. satisfies:
%e x = (A(x)-1)/A(x)^7 + (A(x)-1)*(A(x)^2-1)/A(x)^15 + (A(x)-1)*(A(x)^2-1)*(A(x)^3-1)/A(x)^24 + (A(x)-1)*(A(x)^2-1)*(A(x)^3-1)*(A(x)^4-1)/A(x)^34 + (A(x)-1)*(A(x)^2-1)*(A(x)^3-1)*(A(x)^4-1)*(A(x)^5-1)/A(x)^45 +...
%t 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^6,{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 *)
%t CoefficientList[1+InverseSeries[Series[x - 5*x^2 + 5*x^3 + 30*x^4 - 65*x^5 - 191*x^6 + 378*x^7 + 1557*x^8 + 103*x^9 - 8551*x^10 - 23911*x^11 - 37958*x^12 - 41831*x^13 - 34156*x^14 - 21179*x^15 - 10015*x^16 - 3571*x^17 - 933*x^18 - 169*x^19 - 19*x^20 - x^21, {x, 0, 20}], x],x] (* _Vaclav Kotesovec_, Dec 01 2014 *)
%o (PARI) {a(n)=if(n<0, 0, polcoeff(1 + serreverse(x - 5*x^2 + 5*x^3 + 30*x^4 - 65*x^5 - 191*x^6 + 378*x^7 + 1557*x^8 + 103*x^9 - 8551*x^10 - 23911*x^11 - 37958*x^12 - 41831*x^13 - 34156*x^14 - 21179*x^15 - 10015*x^16 - 3571*x^17 - 933*x^18 - 169*x^19 - 19*x^20 - x^21 +x^2*O(x^n)), n))}
%o (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)^(6*m)*prod(k=1, m, 1-1/Ser(A)^k)), #A-1)); A[n+1]}
%o for(n=0, 25, print1(a(n), ", "))
%Y Cf. A001002, A181997, A181998, A209441.
%K nonn
%O 0,3
%A _Paul D. Hanna_, Apr 08 2012
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