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A239204 Expansion of ((x-1)*sqrt(x^2-6*x+1)-x^2-4*x+1)/(8*x^3). 2
1, 4, 17, 76, 353, 1688, 8257, 41128, 207905, 1063932, 5501073, 28695252, 150827073, 798054000, 4247388417, 22722717328, 122125206977, 659101365812, 3570473750929, 19407758970268, 105820555054241, 578622868877704, 3172136074486337 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

LINKS

Vincenzo Librandi, Table of n, a(n) for n = 0..250

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)

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 *)

PROG

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

(MAGMA) m:=30; R<x>:=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

(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

CROSSREFS

Cf. A085403, A006318.

Sequence in context: A081910 A026773 A081186 * A005572 A202879 A333059

Adjacent sequences:  A239201 A239202 A239203 * A239205 A239206 A239207

KEYWORD

nonn

AUTHOR

Vladimir Kruchinin, Mar 17 2014

STATUS

approved

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Last modified July 12 18:03 EDT 2020. Contains 335666 sequences. (Running on oeis4.)