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A215715
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G.f. satisfies A(x) = (1 + x*A(x)^2) * (1 + x*A(x)^4).
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4
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1, 2, 13, 118, 1242, 14227, 172177, 2165732, 28032668, 370944717, 4995412647, 68239105203, 943278064473, 13169938895473, 185453340189492, 2630813161415976, 37561512615867450, 539336703889993006, 7783290731579783544, 112828761898680983141, 1642222504807143423470
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OFFSET
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0,2
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COMMENTS
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More generally, for fixed parameters p, q, r, and s, if F(x) satisfies:
F(x) = (1 + x^r*F(x)^(p+1)) * (1 + x^(r+s)*F(x)^(p+q+1)), then
F(x) = exp( Sum_{n>=1} x^(n*r)*F(x)^(n*p)/n * [Sum_{k=0..n} C(n,k)^2 * x^(k*s)*F(x)^(k*q)] ).
The radius of convergence of g.f. A(x) is r = 0.06368546004073732405169450... with A(r) = 1.3960637117611795281240000742797488619448782873... where y=A(r) satisfies 6*y^5 + 17*y^4 - 46*y^3 + 16*y^2 + 4*y - 8 = 0.
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LINKS
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FORMULA
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G.f. A(x) satisfies:
(1) A(x) = sqrt( (1/x)*Series_Reversion( x*(1 + sqrt(1 - 4*x*(1+x)^2))^2/(4*(1+x)^2) ) ).
(2) A(x) = exp( Sum_{n>=1} x^n*A(x)^n/n * Sum_{k=0..n} C(n,k)^2 * A(x)^(2*k) ).
(3) A(x) = exp( Sum_{n>=1} x^n*A(x)^(3*n)/n * Sum_{k=0..n} C(n,k)^2 / A(x)^(2*k) ).
The formal inverse of the g.f. A(x) is (sqrt(x^4 + 4*x^3 - 2*x^2 + 1) - (1+x^2))/(2*x^4).
a(n) ~ c*d^n/(sqrt(Pi)*n^(3/2)), where d = 15.70217125479403872... is the root of the equation -1024 - 3840*d + 26368*d^2 - 58644*d^3 + 1933*d^4 + 108*d^5 = 0 and c = 0.320114409... - Vaclav Kotesovec, Sep 17 2013
a(n) = Sum_{k=0..n} binomial(2*n+2*k+1,k) * binomial(2*n+2*k+1,n-k) / (2*n+2*k+1). - Seiichi Manyama, Jul 18 2023
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EXAMPLE
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G.f.: A(x) = 1 + 2*x + 13*x^2 + 118*x^3 + 1242*x^4 + 14227*x^5 + ...
Related expansions.
A(x)^2 = 1 + 4*x + 30*x^2 + 288*x^3 + 3125*x^4 + 36490*x^5 + ...
A(x)^4 = 1 + 8*x + 76*x^2 + 816*x^3 + 9454*x^4 + 115260*x^5 + ...
A(x)^6 = 1 + 12*x + 138*x^2 + 1648*x^3 + 20427*x^4 + 260934*x^5 + ...
where A(x) = 1 + x*(A(x)^2 + A(x)^4) + x^2*A(x)^6.
The logarithm of the g.f. equals the series:
log(A(x)) = (1 + A(x)^2)*x*A(x) + (1 + 2^2*A(x)^2 + A(x)^4)*x^2*A(x)^2/2 +
(1 + 3^2*A(x)^2 + 3^2*A(x)^4 + A(x)^6)*x^3*A(x)^3/3 +
(1 + 4^2*A(x)^2 + 6^2*A(x)^4 + 4^2*A(x)^6 + A(x)^8)*x^4*A(x)^4/4 +
(1 + 5^2*A(x)^2 + 10^2*A(x)^4 + 10^2*A(x)^6 + 5^2*A(x)^8 + A(x)^10)*x^5*A(x)^5/5 + ...
Explicitly,
log(A(x)) = 2*x + 22*x^2/2 + 284*x^3/3 + 3878*x^4/4 + 54607*x^5/5 + 784144*x^6/6 + 11414265*x^7/7 + 167819014*x^8/8 + ...
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MATHEMATICA
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CoefficientList[Sqrt[1/x*InverseSeries[Series[x*(1+Sqrt[1-4*x*(1+x)^2])^2/(4*(1+x)^2), {x, 0, 20}], x]], x] (* Vaclav Kotesovec, Sep 17 2013 *)
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PROG
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(PARI) {a(n)=local(A=1+x+x*O(x^n)); for(i=1, n, A=(1 + x*A^2)*(1 + x*A^4)); polcoeff(A, n)}
for(n=0, 31, print1(a(n), ", "))
(PARI) {a(n)=polcoeff( sqrt((1/x)*serreverse( x*(1 + sqrt(1 - 4*x*(1+x)^2 +x*O(x^n)))^2/(4*(1+x)^2))), 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*A^(2*j))*x^m*A^m/m))); polcoeff(A, 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/A^(2*j))*x^m*A^(3*m)/m))); 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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