A242781 Expansion of (1 - 2*x - sqrt(1-4*x))/(4*x^2 + sqrt(1-4*x)*(3*x+1) - 5*x + 1).

0, 0, 1, 4, 15, 57, 217, 828, 3169, 12165, 46827, 180701, 698867, 2708307, 10514331, 40885356, 159216543, 620845293, 2423825649, 9473195889, 37061983617, 145131715707, 568808493081, 2231063305461, 8757391892965, 34397931629763, 135196161588037, 531682892209431

Offset

0,4

Links

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

Formula

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

G.f.: log'(1/(1-x^3*C(x)^3))/3, where C(x) is g.f. of A000108.

a(n) ~ 2^(2*n+1)/(7*sqrt(Pi*n)). - Vaclav Kotesovec, May 24 2014

Conjecture D-finite with recurrence: 3*(n+1)*a(n) -18*n*a(n-1) +2*(11*n-13)*a(n-2) +6*(n-7)*a(n-3) +(7*n-9)*a(n-4) +2*(2*n-7)*a(n-5)=0. - R. J. Mathar, Jan 25 2020

Mathematica

CoefficientList[Series[(1-2*x-Sqrt[1-4*x])/(4*x^2+Sqrt[1-4*x]*(3*x+1)-5*x+1), {x, 0, 20}], x] (* Vaclav Kotesovec, May 24 2014 *)

Prog

(Maxima)
a(n):=sum(binomial(2*n-3*k+1, n-3*k+1), k, 1, ceiling((n+1)/3));
(PARI) x='x+O('x^50); concat([0, 0], Vec((1-2*x-sqrt(1-4*x))/(4*x^2 + sqrt(1-4*x)*(3*x+1)-5*x+1))) \\ G. C. Greubel, Jun 01 2017

Crossrefs

Cf. A000108.

Sequence in context: A077823 A047108 A125145 * A346195 A371854 A277924

Adjacent sequences: A242778 A242779 A242780 * A242782 A242783 A242784

Keyword

nonn

Author

Vladimir Kruchinin, May 24 2014

Status

approved