cp's OEIS Frontend

A192737 G.f. satisfies: A(x) = 1/(1 - x*A(x)/(1 - x*A(x)^2/(1 - x*A(x)^3/(1 - x*A(x)^4/(1 - ...))))), a recursive continued fraction.

%I A192737 #12 Feb 16 2025 08:33:15
%S A192737 1,1,3,13,68,399,2531,16994,119199,865849,6474177,49616016,388484212,
%T A192737 3100311228,25172981053,207665895001,1738775327319,14764815028481,
%U A192737 127076945426555,1108103873824072,9787004793441886,87539719110388691
%N A192737 G.f. satisfies: A(x) = 1/(1 - x*A(x)/(1 - x*A(x)^2/(1 - x*A(x)^3/(1 - x*A(x)^4/(1 - ...))))), a recursive continued fraction.
%H A192737 Vaclav Kotesovec, <a href="/A192737/b192737.txt">Table of n, a(n) for n = 0..250</a>
%H A192737 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/Rogers-RamanujanContinuedFraction.html">Rogers-Ramanujan Continued Fraction</a>.
%F A192737 G.f. satisfies: A(x) = P(x)/Q(x) where
%F A192737   P(x) = Sum_{n>=0} (-x)^n * A(x)^(n*(n+1)) / Product_{k=1..n} (1-A(x)^k),
%F A192737   Q(x) = Sum_{n>=0} (-x)^n * A(x)^(n^2) / Product_{k=1..n} (1-A(x)^k),
%F A192737 due to Ramanujan's continued fraction identity.
%e A192737 G.f.: A(x) = 1 + x + 3*x^2 + 13*x^3 + 68*x^4 + 399*x^5 + 2531*x^6 +...
%e A192737 The g.f. A = A(x) satisfies A = P(x)/Q(x) where
%e A192737 P(x) = 1 - x*A^2/(1-A) + x^2*A^6/((1-A)*(1-A^2)) - x^3*A^12/((1-A)*(1-A^2)*(1-A^3)) + x^4*A^20/((1-A)*(1-A^2)*(1-A^3)*(1-A^4)) -+...
%e A192737 Q(x) = 1 - x*A/(1-A) + x^2*A^4/((1-A)*(1-A^2)) - x^3*A^9/((1-A)*(1-A^2)*(1-A^3)) + x^4*A^16/((1-A)*(1-A^2)*(1-A^3)*(1-A^4)) -+...
%e A192737 Explicitly, the above series begin:
%e A192737 P(x) = exp(1)*(1 - 1/4*x - 283/288*x^2 - 6223/1152*x^3 - 140734037/4147200*x^4 - 3826874463/16588800*x^5 - 290665690804549/175575859200*x^6 +...);
%e A192737 Q(x) = exp(1)*(1 - 5/4*x - 787/288*x^2 - 13731/1152*x^3 - 271921637/4147200*x^4 - 6765586315/16588800*x^5 - 481505562390493/175575859200*x^6 +...).
%o A192737 (PARI) /* As a recursive continued fraction: */
%o A192737 {a(n)=local(A=1+x, CF); for(i=1, n, CF=1+x; for(k=0, n, CF=1/(1-x*A^(n-k+1)*CF+x*O(x^n))); A=CF); polcoeff(A, n)}
%o A192737 (PARI) /* By Ramanujan's continued fraction identity: */
%o A192737 {a(n)=local(A=1+x, P, Q); for(i=1, n,
%o A192737 P=sum(m=0, 2*n, (-x)^m*A^(m*(m+1))/prod(k=1, m, 1-A^k)/exp(1)+x*O(x^(2*n)));
%o A192737 Q=sum(m=0, 2*n, (-x)^m*A^(m^2)/prod(k=1, m, 1-A^k)/exp(1)+x*O(x^(2*n))); A=round(P/Q)); polcoeff(A, n)}
%Y A192737 Cf. A005169, A192728, A192729, A192730.
%K A192737 nonn
%O A192737 0,3
%A A192737 _Paul D. Hanna_, Jul 08 2011