A071717 Expansion of (1 + x^2*C)*C^2, where C = (1 - sqrt(1-4*x))/(2*x) is g.f. for Catalan numbers, A000108.

1, 2, 6, 17, 51, 160, 519, 1727, 5863, 20228, 70720, 250002, 892126, 3209328, 11626385, 42378075, 155307615, 571925820, 2115257100, 7853744910, 29263124250, 109384710240, 410075910270, 1541481197334, 5808790935126

Offset: 0

Author

N. J. A. Sloane, Jun 06 2002

Keywords

• nonn

Links

• Michael De Vlieger, Table of n, a(n) for n = 0..1000
• Stoyan Dimitrov, On permutation patterns with constrained gap sizes, arXiv:2002.12322 [math.CO], 2020.

Crossrefs

Cf. A000108, A071716, A071718, A071719.

Programs

Maple

seq(coeff(series( ((1-x-3*x^2) -(1+x-x^2)*sqrt(1-4*x))/(2*x^2) , x, n+1), x, n), n = 0..30); # G. C. Greubel, May 30 2020

Mathematica

With[{$MaxExtraPrecision = 1000}, CoefficientList[Series[(1 + x^2*#)*#^2 &[(1 - (1 - 4 x)^(1/2))/(2 x)], {x, 0, 24}], x]] (* Michael De Vlieger, May 30 2020 *)

Sage

def A071717_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( ((1-x-3*x^2) -(1+x-x^2)*sqrt(1-4*x))/(2*x^2) ).list()
A071717_list(30) # G. C. Greubel, May 30 2020

Formula

Conjecture: (n+2)*a(n) +(-3*n-2)*a(n-1) +(-5*n+8)*a(n-2) +2*(2*n-7)*a(n-3)=0. - R. J. Mathar, Aug 25 2013

G.f.: ( (1 -x -3*x^2) - (1 +x -x^2)*sqrt(1-4*x) )/(2*x^2). - G. C. Greubel, May 30 2020