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A215661
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G.f. satisfies: A(x) = (1 + 2*x*A(x)) * (1 + x*A(x)^2).
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2
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1, 3, 14, 83, 554, 3966, 29756, 230915, 1838162, 14926346, 123157572, 1029590062, 8702171620, 74238432924, 638408311800, 5528154378467, 48161687414498, 421848099386322, 3712675503776372, 32815429463428794, 291169073934720940, 2592569269501484836
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OFFSET
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0,2
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COMMENTS
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The radius of convergence of g.f. A(x) is r = 0.10464695509817751113707000... with A(r) = 2.224485325158190991256253303513498621559794760... where y = A(r) satisfies 9*y^3 - 22*y^2 + 8*y - 8 = 0.
r = 1/((2/27*((76276+3348*sqrt(93))^(2/3) + 1684 + 46*(76276 + 3348*sqrt(93))^(1/3))/(76276 + 3348*sqrt(93))^(1/3))). - Vaclav Kotesovec, Sep 16 2013
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LINKS
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FORMULA
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G.f. A(x) satisfies:
(1) A(x) = exp( Sum_{n>=1} x^n*A(x)^n/n * Sum_{k=0..n} C(n,k)^2 * 2^k/A(x)^k ).
(2) A(x) = (1/x) * Series_Reversion( x*(1-x-2*x^2)/(1+2*x) ).
(3) A(x) = Sum_{n>=0} A001045(n+2) * x^n * A(x)^n, where A001045 is the Jacobsthal numbers.
The formal inverse of the g.f. A(x) is (sqrt(4-4*x+9*x^2) - (2+x))/(4*x^2).
a(n) = [x^n] ( (1+2*x)/(1-x-2*x^2) )^(n+1) / (n+1).
Recurrence: 9*n*(n+1)*(31*n-55)*a(n) = 2*n*(1426*n^2 - 3243*n + 1211)*a(n-1) - 8*(248*n^3 - 936*n^2 + 991*n - 240)*a(n-2) + 32*(n-2)*(2*n-5)*(31*n-24)*a(n-3). - Vaclav Kotesovec, Sep 16 2013
a(n) ~ 1/558*sqrt(186)*sqrt((556299836 + 9879948*sqrt(93))^(1/3) * ((556299836 + 9879948*sqrt(93))^(2/3) + 669724 + 806*(556299836 + 9879948*sqrt(93))^(1/3)))/((556299836 + 9879948*sqrt(93))^(1/3)) * (2/27*((76276+3348*sqrt(93))^(2/3) + 1684 + 46*(76276+3348*sqrt(93))^(1/3))/ (76276+3348*sqrt(93))^(1/3))^n / (sqrt(Pi)*n^(3/2)). - Vaclav Kotesovec, Sep 16 2013
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EXAMPLE
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G.f.: A(x) = 1 + 3*x + 14*x^2 + 83*x^3 + 554*x^4 + 3966*x^5 + 29756*x^6 + ...
Related expansions.
A(x)^2 = 1 + 6*x + 37*x^2 + 250*x^3 + 1802*x^4 + 13580*x^5 + 105709*x^6 + ...
A(x)^3 = 1 + 9*x + 69*x^2 + 528*x^3 + 4122*x^4 + 32847*x^5 + ...
where A(x) = 1 + x*(2*A(x) + A(x)^2) + 2*x^2*A(x)^3.
The g.f. also satisfies the series:
A(x) = 1 + 3*x*A(x) + 5*x^2*A(x)^2 + 11*x^3*A(x)^3 + 21*x^4*A(x)^4 + 43*x^5*A(x)^5 + 85*x^6*A(x)^6 + ... + Jacobsthal(n+2)*x^n*A(x)^n + ...
The logarithm of the g.f. equals the series:
log(A(x)) = (1 + 2/A(x))*x*A(x) + (1 + 2^2*2/A(x) + 2^2/A(x)^2)*x^2*A(x)^2/2 +
(1 + 3^2*2/A(x) + 3^2*2^2/A(x)^2 + 2^3/A(x)^3)*x^3*A(x)^3/3 +
(1 + 4^2*2/A(x) + 6^2*2^2/A(x)^2 + 4^2*2^3/A(x)^3 + 2^4/A(x)^4)*x^4*A(x)^4/4 +
(1 + 5^2*2/A(x) + 10^2*2^2/A(x)^2 + 10^2*2^3/A(x)^3 + 5^2*2^4/A(x)^4 + 2^5/A(x)^5)*x^5*A(x)^5/5 + ...
Explicitly,
log(A(x)) = 3*x + 19*x^2/2 + 150*x^3/3 + 1251*x^4/4 + 10738*x^5/5 + 93934*x^6/6 + 832716*x^7/7 + 7454755*x^8/8 + ... + L(n)*x^n/n + ...
where L(n) = [x^n] (1+2*x)^n/(1-x-2*x^2)^n.
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MATHEMATICA
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CoefficientList[1/x * InverseSeries[Series[x*(1-x-2*x^2)/(1+2*x), {x, 0, 20}], x], x] (* Vaclav Kotesovec, Sep 16 2013 *)
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PROG
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(PARI) {a(n)=local(A=1+x); for(i=1, n, A=(1 + 2*x*A)*(1 + x*(A+x*O(x^n))^2)); polcoeff(A, n)}
(PARI) {a(n)=polcoeff( (1/x)*serreverse( x*(1-x-2*x^2)/(1+2*x +x*O(x^n))), n)}
(PARI) {a(n)=local(A=1+x+x*O(x^n)); for(i=1, n, A=exp(sum(m=1, n, sum(j=0, m, binomial(m, j)^2*2^j/A^j)*x^m*A^m/m))); polcoeff(A, n)}
for(n=0, 31, print1(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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