A307490 G.f. A(x) satisfies: A(x) = 1/(1 - x*A(x) - x^2*A(x)^2/(1 - x^2*A(x)^2/(1 - x^2*A(x)^2/(1 - ...)))), a continued fraction.

1, 1, 3, 10, 39, 161, 700, 3144, 14495, 68167, 325787, 1577654, 7724612, 38176620, 190196440, 954182528, 4816268367, 24441691827, 124633707865, 638272397350, 3281390284623, 16929034639153, 87617665434328, 454796186669800, 2367025393846724, 12349738834935876

Offset

0,3

Links

Table of n, a(n) for n=0..25.

Formula

G.f. A(x) satisfies: A(x) = 2/(1 - 2*x*A(x) + sqrt(1 - 4*x^2*A(x)^2)).

G.f. A(x) satisfies: A(x) = Sum_{k>=0} binomial(k,floor(k/2))*x^k*A(x)^k.

G.f.: A(x) = (1/x)*Series_Reversion(x*(1 - 2*x + sqrt(1 - 4*x^2))/2).

a(n) ~ 1/(n^(3/2)*sqrt(Pi*(103/6 + (1/6)*sqrt(36037)*cos((1/3)*(4*Pi + arccos(6832781/(36037*sqrt(36037))))))) * (-7/48 + (1/24)*sqrt(203/2) * cos((1/3)*arccos(-(1849/(203*sqrt(406))))))^n). - Vaclav Kotesovec, Sep 16 2021

Example

G.f.: A(x) = 1 + x + 3*x^2 + 10*x^3 + 39*x^4 + 161*x^5 + 700*x^6 + 3144*x^7 + 14495*x^8 + 68167*x^9 + 325787*x^10 + ...

Mathematica

terms = 26; CoefficientList[1/x InverseSeries[Series[x (1 - 2 x + Sqrt[1 - 4 x^2])/2, {x, 0, terms}], x], x]
terms = 26; A[_] = 0; Do[A[x_] = 1/(1 - x A[x] + ContinuedFractionK[-x^2 A[x]^2, 1, {k, 1, j}]) + O[x]^j, {j, 1, terms}]; CoefficientList[A[x], x]
terms = 25; A[_] = 1; Do[A[x_] = 2/(1 - 2 x A[x] + Sqrt[1 - 4 x^2 A[x]^2]) + O[x]^(terms + 1) // Normal, {terms + 1}]; CoefficientList[A[x], x]
terms = 26; A[_] = 1; Do[A[x_] = Sum[Binomial[k, Floor[k/2]] x^k A[x]^k, {k, 0, j}] + O[x]^j, {j, 1, terms}]; CoefficientList[A[x], x]

Crossrefs

Cf. A001405, A001764, A078531, A307489.

Sequence in context: A371713 A218001 A378247 * A253194 A367258 A338185

Adjacent sequences: A307487 A307488 A307489 * A307491 A307492 A307493

Keyword

nonn

Author

Ilya Gutkovskiy, Apr 10 2019

Status

approved