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%I #17 Dec 01 2014 07:09:37
%S 1,1,2,9,46,259,1539,9484,59961,386319,2524940,16687599,111264335,
%T 747080253,5044629212,34218868880,232964088130,1590660486297,
%U 10885758313976,74627209920879,512254418843196,3519150502675731,24187028454513735,166249089897708930
%N G.f. satisfies: x = Sum_{n>=1} 1/A(x)^(3*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.
%F G.f. satisfies: 1+x = A(y) where y = x - 2*x^2 - x^3 + 4*x^4 + 4*x^5 + x^6.
%F G.f. satisfies: x = Sum_{n>=1} 1/A(x)^(n*(n+7)/2) * Product_{k=1..n} (A(x)^k - 1).
%e G.f.: A(x) = 1 + x + 2*x^2 + 9*x^3 + 46*x^4 + 259*x^5 + 1539*x^6 +...
%e The g.f. satisfies:
%e x = (A(x)-1)/A(x)^4 + (A(x)-1)*(A(x)^2-1)/A(x)^9 + (A(x)-1)*(A(x)^2-1)*(A(x)^3-1)/A(x)^15 + (A(x)-1)*(A(x)^2-1)*(A(x)^3-1)*(A(x)^4-1)/A(x)^22 + (A(x)-1)*(A(x)^2-1)*(A(x)^3-1)*(A(x)^4-1)*(A(x)^5-1)/A(x)^30 +...
%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^3,{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 - 2*x^2 - x^3 + 4*x^4 + 4*x^5 + x^6, {x, 0, 20}], x],x] (* _Vaclav Kotesovec_, Dec 01 2014 *)
%o (PARI) {a(n)=if(n<0,0,polcoeff(1 + serreverse(x - 2*x^2 - x^3 + 4*x^4 + 4*x^5 + x^6 +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)^(3*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, A181998, A209441, A209442, A214693 (variant).
%K nonn
%O 0,3
%A _Paul D. Hanna_, Apr 05 2012
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