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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. 11
1, 1, 2, 6, 19, 64, 226, 822, 3061, 11615, 44746, 174552, 688122, 2737153, 10972066, 44279234, 179754362, 733554695, 3007551211, 12382623614, 51174497023, 212218265661, 882810782322, 3682922292680, 15404800893438, 64590512696020, 271425803359505 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

LINKS

Vaclav Kotesovec, Table of n, a(n) for n = 0..400

Eric Weisstein's World of Mathematics, Rogers-Ramanujan Continued Fraction.

FORMULA

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

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

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

due to Ramanujan's continued fraction identity.

a(n) ~ c * d^n / n^(3/2), where d = 4.44776682810490219629673157389741... and c = 0.533241700941579126635423052024... - Vaclav Kotesovec, Apr 30 2017

EXAMPLE

G.f.: A(x) = 1 + x + 2*x^2 + 6*x^3 + 19*x^4 + 64*x^5 + 226*x^6 +...

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

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)) -+...

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)) -+...

Explicitly, the above series begin:

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 +...

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 +...

Also, the g.f. A = A(x) satisfies:

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) +...

which is a series generated by the continued fraction expression.

PROG

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

{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)}

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

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

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

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)}

CROSSREFS

Cf. A005169, A192729, A192730.

Sequence in context: A176950 A119370 A192738 * A181315 A181734 A216447

Adjacent sequences:  A192725 A192726 A192727 * A192729 A192730 A192731

KEYWORD

nonn

AUTHOR

Paul D. Hanna, Jul 08 2011

STATUS

approved

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Last modified June 1 09:53 EDT 2020. Contains 334762 sequences. (Running on oeis4.)