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A181998 G.f. satisfies: x = Sum_{n>=1} 1/A(x)^(4*n) * Product_{k=1..n} (1 - 1/A(x)^k). 6

%I

%S 1,1,3,18,124,935,7443,61510,522467,4532452,39985628,357641094,

%T 3235846003,29565353095,272429349163,2528938553028,23629834081955,

%U 222080711420655,2098112946860819,19915641133236764,189853287434733709,1816924035668823659,17450483777418686431

%N G.f. satisfies: x = Sum_{n>=1} 1/A(x)^(4*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 - 3*x^2 + 11*x^4 + x^5 - 30*x^6 - 42*x^7 - 26*x^8 - 8*x^9 - x^10.

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

%e G.f.: A(x) = 1 + x + 3*x^2 + 18*x^3 + 124*x^4 + 935*x^5 + 7443*x^6 +...

%e The g.f. satisfies:

%e 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 +...

%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^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 *)

%t 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 *)

%o (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))}

%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)^(4*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, A209441, A209442, A214694 (variant).

%K nonn

%O 0,3

%A _Paul D. Hanna_, Apr 05 2012

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Last modified June 3 05:29 EDT 2020. Contains 334798 sequences. (Running on oeis4.)