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A193021
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G.f.: A(x) = 1/(1 - x*(1+x)/(1 - x^2*(1+x)/(1 - x^3*(1+x)/(1 - x^4*(1+x)/(1 - ...))))), a continued fraction.
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0
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1, 1, 2, 4, 9, 19, 42, 93, 205, 453, 1003, 2221, 4918, 10892, 24126, 53442, 118384, 262248, 580946, 1286953, 2850965, 6315712, 13991153, 30994539, 68662111, 152107038, 336962513, 746472721, 1653660451, 3663352982, 8115423952, 17978094917
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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.: A(x) = P(x)/Q(x) where
_ P(x) = Sum_{n>=0} x^(n*(n+1)) * (-1-x)^n / Product(k=1..n} (1-x^k),
_ Q(x) = Sum_{n>=0} x^(n^2) * (-1-x)^n / Product(k=1..n} (1-x^k),
due to Ramanujan's continued fraction identity.
a(n) ~ c * d^n, where d = 2.2152996327586394990264647692065917932114805328469811... and c = 0.35183326334197478337804661003215013650248042019243949..., d = 1/r, where r = 0.451406204927110926742870001969351624948735236137... is the lowest root of the equation Q(r) = 0. - Vaclav Kotesovec, Aug 25 2017, updated Sep 24 2020
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EXAMPLE
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G.f.: A(x) = 1 + x + 2*x^2 + 4*x^3 + 9*x^4 + 19*x^5 + 42*x^6 +...
which satisfies A(x) = P(x)/Q(x) where
P(x) = 1 - x^2*(1+x)/(1-x) + x^6*(1+x)^2/((1-x)*(1-x^2)) - x^12*(1+x)^3/((1-x)*(1-x^2)*(1-x^3)) + x^20*(1+x)^4/((1-x)*(1-x^2)*(1-x^3)*(1-x^4)) +...
Q(x) = 1 - x*(1+x)/(1-x) + x^4*(1+x)^2/((1-x)*(1-x^2)) - x^9*(1+x)^3/((1-x)*(1-x^2)*(1-x^3)) + x^16*(1+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 - 2*x^4 - 2*x^5 - x^6 + x^7 + 3*x^8 + 5*x^9 + 7*x^10 + 9*x^11 + 10*x^12 + 9*x^13 + 7*x^14 + 4*x^15 - x^16 - 7*x^17 - 14*x^18 +...
Q(x) = 1 - x - 2*x^2 - 2*x^3 - x^4 + x^5 + 3*x^6 + 5*x^7 + 7*x^8 + 8*x^9 + 7*x^10 + 5*x^11 + 2*x^12 - 3*x^13 - 9*x^14 - 16*x^15 - 24*x^16 - 30*x^17 +...
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MATHEMATICA
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nmax = 50; CoefficientList[Series[1/Fold[(1 - #2/#1) &, 1, Reverse[x^Range[nmax + 1]*(1+x)]], {x, 0, nmax}], x] (* Vaclav Kotesovec, Aug 25 2017 *)
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PROG
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(PARI) /* As a continued fraction: */
{a(n)=local(A=1+x, CF); CF=1+x; for(k=0, n, CF=1/(1-x^(n-k+1)*(1+x)*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)*(-1-x+x*O(x^n))^m);
Q=sum(m=0, sqrtint(n), x^(m^2)/prod(k=1, m, 1-x^k)*(-1-x+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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