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G.f. satisfies: A(x) = 1/(1 - x*A(x)^2/(1 - x^2*A(x)^2/(1 - x^3*A(x)^2/(1 - x^4*A(x)^2/(1 - ...))))), a recursive continued fraction.
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%I #9 Aug 25 2017 08:11:06

%S 1,1,3,13,63,329,1808,10299,60271,360198,2189111,13488379,84066176,

%T 529037390,3357014851,21455604032,137993279809,892448240335,

%U 5800266701499,37864046563210,248158092634265,1632254493141021,10771183395497445

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

%H Vaclav Kotesovec, <a href="/A192729/b192729.txt">Table of n, a(n) for n = 0..300</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/Rogers-RamanujanContinuedFraction.html">Rogers-Ramanujan Continued Fraction</a>.

%F G.f. satisfies: A(x) = P(x)/Q(x) where

%F _ P(x) = Sum_{n>=0} x^(n*(n+1)) * (-A(x)^2)^n / Product(k=1..n} (1-x^k),

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

%F due to Ramanujan's continued fraction identity.

%F a(n) ~ c * d^n / n^(3/2), where d = 7.0656326355634513691927118582399... and c = 0.2386935555822482686868972746... - _Vaclav Kotesovec_, Aug 25 2017

%e G.f.: A(x) = 1 + x + 3*x^2 + 13*x^3 + 63*x^4 + 329*x^5 + 1808*x^6 +...

%e which satisfies A(x) = P(x)/Q(x) where

%e P(x) = 1 - x^2*A(x)^2/(1-x) + x^6*A(x)^4/((1-x)*(1-x^2)) - x^12*A(x)^6/((1-x)*(1-x^2)*(1-x^3)) + x^20*A(x)^8/((1-x)*(1-x^2)*(1-x^3)*(1-x^4)) -+...

%e Q(x) = 1 - x*A(x)^2/(1-x) + x^4*A(x)^4/((1-x)*(1-x^2)) - x^9*A(x)^6/((1-x)*(1-x^2)*(1-x^3)) + x^16*A(x)^8/((1-x)*(1-x^2)*(1-x^3)*(1-x^4)) -+...

%e Explicitly, the above series begin:

%e P(x) = 1 - x^2 - 3*x^3 - 10*x^4 - 42*x^5 - 202*x^6 - 1060*x^7 - 5862*x^8 - 33592*x^9 - 197585*x^10 - 1185867*x^11 - 7233049*x^12 +...

%e Q(x) = 1 - x - 3*x^2 - 10*x^3 - 41*x^4 - 198*x^5 - 1041*x^6 - 5766*x^7 - 33074*x^8 - 194674*x^9 - 1168988*x^10 - 7132869*x^11 - 44097821*x^12 +...

%o (PARI) /* As a recursive continued fraction: */

%o {a(n)=local(A=1+x,CF);for(i=1,n,CF=1+x;for(k=0,n,CF=1/(1-x^(n-k+1)*A^2*CF+x*O(x^n)));A=CF);polcoeff(A,n)}

%o (PARI) /* By Ramanujan's continued fraction identity: */

%o {a(n)=local(A=1+x,P,Q);for(i=1,n,

%o P=sum(m=0,sqrtint(n),x^(m*(m+1))/prod(k=1,m,1-x^k)*(-A^2+x*O(x^n))^m);

%o Q=sum(m=0,sqrtint(n),x^(m^2)/prod(k=1,m,1-x^k)*(-A^2+x*O(x^n))^m);A=P/Q);polcoeff(A,n)}

%Y Cf. A005169, A192728, A192730.

%K nonn

%O 0,3

%A _Paul D. Hanna_, Jul 08 2011