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A192728 G.f. satisfies: A(x) = 1/(1 - x*A(x)/(1 - x^2*A(x)/(1 - x^3*A(x)/(1 - x^4*A(x)/(1 - ...))))), a recursive continued fraction.

%I A192728 #17 Feb 16 2025 08:33:15
%S A192728 1,1,2,6,19,64,226,822,3061,11615,44746,174552,688122,2737153,
%T A192728 10972066,44279234,179754362,733554695,3007551211,12382623614,
%U A192728 51174497023,212218265661,882810782322,3682922292680,15404800893438,64590512696020,271425803359505
%N A192728 G.f. satisfies: A(x) = 1/(1 - x*A(x)/(1 - x^2*A(x)/(1 - x^3*A(x)/(1 - x^4*A(x)/(1 - ...))))), a recursive continued fraction.
%H A192728 Vaclav Kotesovec, <a href="/A192728/b192728.txt">Table of n, a(n) for n = 0..400</a>
%H A192728 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/Rogers-RamanujanContinuedFraction.html">Rogers-Ramanujan Continued Fraction</a>.
%F A192728 G.f. satisfies: A(x) = P(x)/Q(x) where
%F A192728   P(x) = Sum_{n>=0} x^(n*(n+1)) * (-A(x))^n / Product_{k=1..n} (1-x^k),
%F A192728   Q(x) = Sum_{n>=0} x^(n^2) * (-A(x))^n / Product_{k=1..n} (1-x^k),
%F A192728 due to Ramanujan's continued fraction identity.
%F A192728 a(n) ~ c * d^n / n^(3/2), where d = 4.44776682810490219629673157389741... and c = 0.533241700941579126635423052024... - _Vaclav Kotesovec_, Apr 30 2017
%e A192728 G.f.: A(x) = 1 + x + 2*x^2 + 6*x^3 + 19*x^4 + 64*x^5 + 226*x^6 +...
%e A192728 which satisfies A(x) = P(x)/Q(x) where
%e A192728 P(x) = 1 - x^2*A(x)/(1-x) + x^6*A(x)^2/((1-x)*(1-x^2)) - x^12*A(x)^3/((1-x)*(1-x^2)*(1-x^3)) + x^20*A(x)^4/((1-x)*(1-x^2)*(1-x^3)*(1-x^4)) -+...
%e A192728 Q(x) = 1 - x*A(x)/(1-x) + x^4*A(x)^2/((1-x)*(1-x^2)) - x^9*A(x)^3/((1-x)*(1-x^2)*(1-x^3)) + x^16*A(x)^4/((1-x)*(1-x^2)*(1-x^3)*(1-x^4)) -+...
%e A192728 Explicitly, the above series begin:
%e A192728 P(x) = 1 - x^2 - 2*x^3 - 4*x^4 - 10*x^5 - 28*x^6 - 90*x^7 - 310*x^8 - 1114*x^9 - 4115*x^10 - 15522*x^11 - 59517*x^12 - 231284*x^13 +...
%e A192728 Q(x) = 1 - x - 2*x^2 - 4*x^3 - 9*x^4 - 26*x^5 - 84*x^6 - 292*x^7 - 1054*x^8 - 3908*x^9 - 14774*x^10 - 56742*x^11 - 220778*x^12 - 868452*x^13 +...
%e A192728 Also, the g.f. A = A(x) satisfies:
%e A192728 A = 1 + x*A + x^2*A^2 + x^3*(A^3 + A^2) + x^4*(A^4 + 2*A^3) + x^5*(A^5 + 3*A^4 + A^3) + x^6*(A^6 + 4*A^5 + 3*A^4 + A^3) + x^7*(A^7 + 5*A^6 + 6*A^5 + 3*A^4) +...
%e A192728 which is a series generated by the continued fraction expression.
%o A192728 (PARI) /* As a recursive continued fraction: */
%o A192728 {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*CF+x*O(x^n)));A=CF);polcoeff(A,n)}
%o A192728 (PARI) /* By Ramanujan's continued fraction identity: */
%o A192728 {a(n)=local(A=1+x,P,Q);for(i=1,n,
%o A192728 P=sum(m=0,sqrtint(n),x^(m*(m+1))/prod(k=1,m,1-x^k)*(-A+x*O(x^n))^m);
%o A192728 Q=sum(m=0,sqrtint(n),x^(m^2)/prod(k=1,m,1-x^k)*(-A+x*O(x^n))^m);A=P/Q);polcoeff(A,n)}
%Y A192728 Cf. A005169, A192729, A192730.
%K A192728 nonn
%O A192728 0,3
%A A192728 _Paul D. Hanna_, Jul 08 2011