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, 3, 19, 152, 1367, 13195, 133556, 1398696, 15029311, 164764985, 1835614027, 20722066612, 236524088612, 2725081792932, 31649837891768, 370161223462480, 4355751419996559, 51532214460643957, 612604251998847641, 7313900470316335280, 87659840436181657215

Offset

0,2

Links

Table of n, a(n) for n=0..20.

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

Cf. A001700, A001764, A078531, A246432, A307490.

Sequence in context: A007111 A346385 A113013 * A323668 A235134 A235322

Adjacent sequences: A307486 A307487 A307488 * A307490 A307491 A307492

Keyword

nonn

Author

Ilya Gutkovskiy, Apr 10 2019

Status

approved