cp's OEIS Frontend

A209441 G.f. satisfies: x = Sum_{n>=1} 1/A(x)^(5*n) * Product_{k=1..n} (1 - 1/A(x)^k).

%I A209441 #11 Jan 03 2018 17:53:45
%S A209441 1,1,4,30,260,2463,24656,256493,2745149,30031677,334334789,3775539592,
%T A209441 43145236171,498018527632,5798165437701,68009060597311,
%U A209441 802908842472516,9533509909631074,113774810189434083,1363985826416978416,16418865502303963429,198369001060550654651
%N A209441 G.f. satisfies: x = Sum_{n>=1} 1/A(x)^(5*n) * Product_{k=1..n} (1 - 1/A(x)^k).
%C A209441 Compare the g.f. to the identity:
%C A209441 G(x) = Sum_{n>=0} 1/G(x)^n * Product_{k=1..n} (1 - 1/G(x)^k)
%C A209441 which holds for all power series G(x) such that G(0)=1.
%H A209441 G. C. Greubel, <a href="/A209441/b209441.txt">Table of n, a(n) for n = 0..900</a>
%F A209441 G.f. satisfies: 1+x = A(y) where y = x - 4*x^2 + 2*x^3 + 20*x^4 - 19*x^5 - 100*x^6 + 3*x^7 + 403*x^8 + 808*x^9 + 861*x^10 + 584*x^11 + 262*x^12 + 76*x^13 + 13*x^14 + x^15.
%F A209441 G.f. satisfies: x = Sum_{n>=1} 1/A(x)^(n*(n+11)/2) * Product_{k=1..n} (A(x)^k - 1).
%e A209441 G.f.: A(x) = 1 + x + 4*x^2 + 30*x^3 + 260*x^4 + 2463*x^5 + 24656*x^6 +...
%e A209441 The g.f. satisfies:
%e A209441 x = (A(x)-1)/A(x)^6 + (A(x)-1)*(A(x)^2-1)/A(x)^13 + (A(x)-1)*(A(x)^2-1)*(A(x)^3-1)/A(x)^21 + (A(x)-1)*(A(x)^2-1)*(A(x)^3-1)*(A(x)^4-1)/A(x)^30 + (A(x)-1)*(A(x)^2-1)*(A(x)^3-1)*(A(x)^4-1)*(A(x)^5-1)/A(x)^40 +...
%t A209441 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^5,{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 A209441 CoefficientList[1+InverseSeries[Series[x - 4*x^2 + 2*x^3 + 20*x^4 - 19*x^5 - 100*x^6 + 3*x^7 + 403*x^8 + 808*x^9 + 861*x^10 + 584*x^11 + 262*x^12 + 76*x^13 + 13*x^14 + x^15, {x, 0, 20}], x],x] (* _Vaclav Kotesovec_, Dec 01 2014 *)
%o A209441 (PARI) {a(n)=if(n<0, 0, polcoeff(1 + serreverse(x - 4*x^2 + 2*x^3 + 20*x^4 - 19*x^5 - 100*x^6 + 3*x^7 + 403*x^8 + 808*x^9 + 861*x^10 + 584*x^11 + 262*x^12 + 76*x^13 + 13*x^14 + x^15 +x^2*O(x^n)), n))}
%o A209441 (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)^(5*m)*prod(k=1, m, 1-1/Ser(A)^k)), #A-1)); A[n+1]}
%o A209441 for(n=0, 25, print1(a(n), ", "))
%Y A209441 Cf. A001002, A181997, A181998, A209442, A214695 (variant).
%K A209441 nonn
%O A209441 0,3
%A A209441 _Paul D. Hanna_, Apr 08 2012