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A307489
G.f. A(x) satisfies: A(x) = 1/(1 - 2*x*A(x) - x*A(x)/(1 - x*A(x)/(1 - x*A(x)/(1 - ...)))), a continued fraction.
1
1, 3, 19, 152, 1367, 13195, 133556, 1398696, 15029311, 164764985, 1835614027, 20722066612, 236524088612, 2725081792932, 31649837891768, 370161223462480, 4355751419996559, 51532214460643957, 612604251998847641, 7313900470316335280, 87659840436181657215
OFFSET
0,2
FORMULA
G.f. A(x) satisfies: A(x) = 2/(1 - 4*x*A(x) + sqrt(1 - 4*x*A(x))).
G.f. A(x) satisfies: A(x) = Sum_{k>=0} binomial(2*k+1,k+1)*x^k*A(x)^k.
G.f.: A(x) = (1/x)*Series_Reversion(x*(1 - 4*x + sqrt(1 - 4*x))/2).
a(n) ~ 2^(12*n + 5) / (sqrt(Pi) * 17^(1/4) * n^(3/2) * (107 + 51*sqrt(17))^(n + 1/2)). - Vaclav Kotesovec, Sep 16 2021
EXAMPLE
G.f.: A(x) = 1 + 3*x + 19*x^2 + 152*x^3 + 1367*x^4 + 13195*x^5 + 133556*x^6 + 1398696*x^7 + 15029311*x^8 + 164764985*x^9 + 1835614027*x^10 + ...
MATHEMATICA
terms = 21; CoefficientList[1/x InverseSeries[Series[x (1 - 4 x + Sqrt[1 - 4 x])/2, {x, 0, terms}], x], x]
terms = 21; A[_] = 0; Do[A[x_] = 1/(1 - 2 x A[x] + ContinuedFractionK[-x A[x], 1, {k, 1, j}]) + O[x]^j, {j, 1, terms}]; CoefficientList[A[x], x]
terms = 20; A[_] = 1; Do[A[x_] = 2/(1 - 4 x A[x] + Sqrt[1 - 4 x A[x]]) + O[x]^(terms + 1) // Normal, {terms + 1}]; CoefficientList[A[x], x]
terms = 21; A[_] = 1; Do[A[x_] = Sum[Binomial[2 k + 1, k + 1] x^k A[x]^k, {k, 0, j}] + O[x]^j, {j, 1, terms}]; CoefficientList[A[x], x]
CROSSREFS
KEYWORD
nonn
AUTHOR
Ilya Gutkovskiy, Apr 10 2019
STATUS
approved