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A192729
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G.f. satisfies: A(x) = 1/(1 - x*A(x)^2/(1 - x^2*A(x)^2/(1 - x^3*A(x)^2/(1 - x^4*A(x)^2/(1 - ...))))), a recursive continued fraction.
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5
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1, 1, 3, 13, 63, 329, 1808, 10299, 60271, 360198, 2189111, 13488379, 84066176, 529037390, 3357014851, 21455604032, 137993279809, 892448240335, 5800266701499, 37864046563210, 248158092634265, 1632254493141021, 10771183395497445
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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)^2)^n / Product(k=1..n} (1-x^k),
_ Q(x) = Sum_{n>=0} x^(n^2) * (-A(x)^2)^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 = 7.0656326355634513691927118582399... and c = 0.2386935555822482686868972746... - Vaclav Kotesovec, Aug 25 2017
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EXAMPLE
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G.f.: A(x) = 1 + x + 3*x^2 + 13*x^3 + 63*x^4 + 329*x^5 + 1808*x^6 +...
which satisfies A(x) = P(x)/Q(x) where
P(x) = 1 - x^2*A(x)^2/(1-x) + x^6*A(x)^4/((1-x)*(1-x^2)) - x^12*A(x)^6/((1-x)*(1-x^2)*(1-x^3)) + x^20*A(x)^8/((1-x)*(1-x^2)*(1-x^3)*(1-x^4)) -+...
Q(x) = 1 - x*A(x)^2/(1-x) + x^4*A(x)^4/((1-x)*(1-x^2)) - x^9*A(x)^6/((1-x)*(1-x^2)*(1-x^3)) + x^16*A(x)^8/((1-x)*(1-x^2)*(1-x^3)*(1-x^4)) -+...
Explicitly, the above series begin:
P(x) = 1 - x^2 - 3*x^3 - 10*x^4 - 42*x^5 - 202*x^6 - 1060*x^7 - 5862*x^8 - 33592*x^9 - 197585*x^10 - 1185867*x^11 - 7233049*x^12 +...
Q(x) = 1 - x - 3*x^2 - 10*x^3 - 41*x^4 - 198*x^5 - 1041*x^6 - 5766*x^7 - 33074*x^8 - 194674*x^9 - 1168988*x^10 - 7132869*x^11 - 44097821*x^12 +...
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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^2*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^2+x*O(x^n))^m);
Q=sum(m=0, sqrtint(n), x^(m^2)/prod(k=1, m, 1-x^k)*(-A^2+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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