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.
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
Offset: 0
Keywords
Examples
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.
Links
- Vaclav Kotesovec, Table of n, a(n) for n = 0..400
- Eric Weisstein's World of Mathematics, Rogers-Ramanujan Continued Fraction.
Programs
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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)}
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