A239204 Expansion of ((x-1)*sqrt(x^2-6*x+1)-x^2-4*x+1)/(8*x^3).

1, 4, 17, 76, 353, 1688, 8257, 41128, 207905, 1063932, 5501073, 28695252, 150827073, 798054000, 4247388417, 22722717328, 122125206977, 659101365812, 3570473750929, 19407758970268, 105820555054241, 578622868877704, 3172136074486337

Offset: 0

Author

Vladimir Kruchinin, Mar 17 2014

Keywords

• nonn

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..250
• Letong Hong and Rupert Li, Length-Four Pattern Avoidance in Inversion Sequences, arXiv:2112.15081 [math.CO], 2021.

Crossrefs

Cf. A085403, A006318.

Programs

Magma

m:=30; R:=LaurentSeriesRing(RationalField(), m); Coefficients(R!(((x-1)*Sqrt(x^2-6*x+1)-x^2-4*x+1)/(8*x^3))); // Bruno Berselli, Mar 18 2014

Mathematica

CoefficientList[Series[((x-1)*Sqrt[x^2-6*x+1]-x^2-4*x+1)/(8*x^3), {x, 0, 20}], x] (* Vaclav Kotesovec, Mar 20 2014 *)

Maxima

a(n):=sum(binomial(n+1, n-i+3)*binomial(n+i,n),i,0,n+3)/(2*(n+1));

Sage

A239204 = lambda n: hypergeometric([-n,-n-2],[2],2)
[Integer(A239204(n).n(100)) for n in range(22)] # Peter Luschny, Sep 23 2014

Formula

a(n) = (Sum_{i=0..n+3} binomial(n+1, n-i+3)*binomial(n+i, n) )/(2*(n+1)).

a(n) ~ sqrt(3*sqrt(2)-4) / (4*sqrt(Pi) * n^(3/2) * (sqrt(2)-1)^(2*n+5)). - Vaclav Kotesovec, Mar 20 2014

a(n) = hypergeom([-n,-n-2], [2], 2). - Peter Luschny, Sep 23 2014

a(n) = (A001003(n+2) - A001003(n+1))/2. - Philippe Deléham, Sep 23 2014

Conjectured to be D-finite with recurrence: (n+3)*a(n) + (-7*n-10)*a(n-1) + (7*n-3)*a(n-2) + (-n+2)*a(n-3) = 0. - R. J. Mathar, Nov 02 2014

From Peter Bala, Jan 28 2020: (Start)

O.g.f. A(x) satisfies A(x*(1 - x)/(1 + x)) = Sum_{n >= 0} (n + 1)^2*x^n.

Equivalently, A(x) = (1 + x*S(x))/(1 - x*S(x))^3 where S(x) is the o.g.f. for the large Schröder numbers A006318. (End)