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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
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
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
KEYWORD
nonn
AUTHOR
N. J. A. Sloane, Jun 06 2002
STATUS
approved