A243255 Expansion of 1/2 - (1/2)*(sqrt((2*sqrt(-3*x^2 - 2*x + 1) + 3*x - 2)/x)).

0, 1, 2, 6, 20, 73, 283, 1149, 4833, 20901, 92406, 415878, 1899153, 8778193, 40988834, 193056506, 916091727, 4375298013, 21015823762, 101454449878, 491976470648, 2395332765741, 11704830457167

Offset

0,3

Links

G. C. Greubel, Table of n, a(n) for n = 0..1000

Formula

a(n) = Sum_{k=0..n-1} (binomial(2*k,k)*Sum_{j=0..n} (binomial(j,-n-k+2*j-1)*binomial(n,j)))/n, n>0, a(0)=0.

G.f.: A(x) = x*M(x)*C(x*M(x)), where M(x) is the g.f. of A001006, and C(x) is the g.f. of A000108.

A(x) satisfies A(x)=x*(1+A(x)-2*A(x)^3+A(x)^4)/(1-A(x)).

a(n) ~ 3^n * 7^(n+1/2) / (sqrt(5*Pi) * n^(3/2) * 4^(n+1)). - Vaclav Kotesovec, Jun 02 2014

Conjecture D-finite with recurrence: 2*n*(2*n+1)*a(n) + (-65*n^2+119*n-42)*a(n-1) + 3*(97*n^2-373*n+342)*a(n-2) - 9*(23*n-68)*(n-3)*a(n-3) - 567*(n-3)*(n-4)*a(n-4) = 0. - R. J. Mathar, Jul 15 2017

Mathematica

CoefficientList[Series[1/2 - (Sqrt[(2*Sqrt[-3*x^2-2*x+1]+3*x-2)/x])/2, {x, 0, 20}], x] (* Vaclav Kotesovec, Jun 02 2014 *)

Prog

(Maxima)
a(n):=sum(binomial(2*k, k)*sum(binomial(j, -n-k+2*j-1)*binomial(n, j), j, 0, n), k, 0, n-1)/n;
(PARI) x='x+O('x^30); concat([0], Vec(1/2-1/2*(sqrt((2*sqrt(-3*x^2 -2*x+1)+3*x-2)/x)))) \\ G. C. Greubel, Oct 06 2018
(Magma) m:=30; R<x>:=PowerSeriesRing(Rationals(), m); [0] cat Coefficients(R!(1/2-1/2*(Sqrt((2*Sqrt(-3*x^2-2*x+1)+3*x-2)/x)))); // G. C. Greubel, Oct 06 2018

Crossrefs

Cf. A000108, A001006.

Sequence in context: A150142 A150143 A150144 * A150145 A150146 A150147

Adjacent sequences: A243252 A243253 A243254 * A243256 A243257 A243258

Keyword

nonn

Author

Vladimir Kruchinin, Jun 01 2014

Status

approved