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A192728
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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.
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11
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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
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OFFSET
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0,3
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LINKS
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FORMULA
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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
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EXAMPLE
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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.
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PROG
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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)}
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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