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G.f. A(x) satisfies: 1 = A(x) - x*A(x)/(A(x) - x*A(x)^2/(A(x) - x*A(x)^3/(A(x) - x*A(x)^4/(A(x) - x*A(x)^5/(A(x) - ...))))), a continued fraction relation.
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%I #18 Nov 07 2020 01:17:05

%S 1,1,1,2,6,22,92,423,2093,10994,60744,350743,2106422,13110304,

%T 84330164,559367278,3819233961,26802388190,193080823079,1426252354150,

%U 10792528835886,83587157097544,662060553448763,5358900630188358,44296806348364981

%N G.f. A(x) satisfies: 1 = A(x) - x*A(x)/(A(x) - x*A(x)^2/(A(x) - x*A(x)^3/(A(x) - x*A(x)^4/(A(x) - x*A(x)^5/(A(x) - ...))))), a continued fraction relation.

%C Note that the continued fraction relation: 1 = F(x) - x*F(x)^k/(F(x) - x*F(x)^k/(F(x) - x*F(x)^k/(F(x) - ...))) holds when F(x) = 1 + x*F(x)^k for a fixed parameter k; this sequence explores the case where the parameter k varies over the positive integers in the continued fraction expression.

%H Paul D. Hanna, <a href="/A338748/b338748.txt">Table of n, a(n) for n = 0..300</a>

%F G.f. A(x) satisfies:

%F (1) A(x) = P(x)/Q(x), where

%F P(x) = Sum_{n>=0} A(x)^(n*(n-1)) * x^n / Product_{k=1..n} (A(x)^k - 1),

%F Q(x) = Sum_{n>=0} A(x)^(n*(n-2)) * x^n / Product_{k=1..n} (A(x)^k - 1),

%F due to Ramanujan's continued fraction identity.

%F (2) A(x) = 1 + x*N(x)/P(x), where

%F N(x) = Sum_{n>=0} A(x)^(n^2) * x^n / Product_{k=1..n} (A(x)^k - 1),

%F P(x) = Sum_{n>=0} A(x)^(n*(n-1)) * x^n / Product_{k=1..n} (A(x)^k - 1).

%F (3) A(x) = x/(1 - R(x)/Q(x)), where

%F Q(x) = Sum_{n>=0} A(x)^(n*(n-2)) * x^n / Product_{k=1..n} (A(x)^k - 1),

%F R(x) = Sum_{n>=0} A(x)^(n*(n-3)) * x^n / Product_{k=1..n} (A(x)^k - 1).

%F (4) A(x) = 1 + x/(1 - x*M(x)/N(x)), where

%F M(x) = Sum_{n>=0} A(x)^(n*(n+1)) * x^n / Product_{k=1..n} (A(x)^k - 1),

%F N(x) = Sum_{n>=0} A(x)^(n^2) * x^n / Product_{k=1..n} (A(x)^k - 1).

%F (5) A(x) = B(x*A(x)) where B(x) = A(x/B(x)) is the g.f. of A338747.

%F (6) A(x) = (1/x)*Series_Reversion( x/B(x) ) where B(x) is the g.f. of A338747.

%e G.f.: A(x) = 1 + x + x^2 + 2*x^3 + 6*x^4 + 22*x^5 + 92*x^6 + 423*x^7 + 2093*x^8 + 10994*x^9 + 60744*x^10 + 350743*x^11 + 2106422*x^12 + ...

%e where

%e 1 = A(x) - x*A(x)/(A(x) - x*A(x)^2/(A(x) - x*A(x)^3/(A(x) - x*A(x)^4/(A(x) - x*A(x)^5/(A(x) - x*A(x)^6/(A(x) - ...)))))).

%e RELATED SERIES.

%e Given B(x) is the g.f. of A338747:

%e B(x) = 1 + x + x^3 + x^4 + 6*x^5 + 17*x^6 + 79*x^7 + 330*x^8 + 1594*x^9 + 7876*x^10 + 41433*x^11 + 226617*x^12 + ...

%e then B(x) = A(x/B(x)) and A(x) = B(x*A(x))

%e where

%e 1 = B(x) - x/(B(x) - x*B(x)/(B(x) - x*B(x)^2/(B(x) - x*B(x)^3/(B(x) - x*B(x)^4/(B(x) - x*B(x)^5/(B(x) - ...)))))).

%o (PARI) {a(n) = my(A=[1], CF=1); for(i=1, n, A=concat(A, 0); for(i=1, #A, CF = Ser(A) - (Ser(A)^(#A-i+1)*x)/CF ); A[#A] = -polcoeff(CF, #A-1) ); H=Ser(A);A[n+1]}

%o for(n=0, 30, print1(a(n), ", "))

%Y Cf. A338747, A338752.

%K nonn

%O 0,4

%A _Paul D. Hanna_, Nov 06 2020