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

0,2

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.

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

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 *)

Prog

(Sage)
def A071717_list(prec):
    P.<x> = 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

Crossrefs

Cf. A000108, A071716, A071718, A071719.

Sequence in context: A153773 A059398 A157002 * A181665 A186239 A148451

Adjacent sequences: A071714 A071715 A071716 * A071718 A071719 A071720

Keyword

nonn

Author

N. J. A. Sloane, Jun 06 2002

Status

approved