A263841 Expansion of (1 - 2*x - x^2)/(sqrt(1+x)*(1-3*x)^(3/2)*2*x) - 1/(2*x).

1, 3, 9, 28, 87, 271, 843, 2619, 8123, 25153, 77763, 240054, 740017, 2278329, 7006093, 21520872, 66039651, 202462113, 620164491, 1898109900, 5805127269, 17741909157, 54188530641, 165405964227, 504601360389, 1538559689751, 4688812503053, 14282580916834, 43486805133903

Offset: 0

Author

N. J. A. Sloane, Nov 02 2015

Keywords

• nonn

Links

• A. Asinowski and G. Rote, Point sets with many non-crossing matchings, arXiv preprint arXiv:1502.04925 [cs.CG], 2015. See Table 1.

Crossrefs

Cf. A005773, A132894, A189911.

Programs

Maple

A263841 := n -> add((k+1)*binomial(n, k)*binomial(n-k, iquo(n-k,2)), k = 0 .. n):
seq(A263841(n), n = 0 .. 28); # Mélika Tebni, Jan 25 2024

Mathematica

CoefficientList[Series[(1-2x-x^2)/(Sqrt[1+x] (1-3x)^(3/2) 2x)-1/(2x),{x,0,30}],x] (* Harvey P. Dale, Aug 21 2017 *)

PARI

my(x='x+O('x^40)); Vec((1-2*x-x^2)/(sqrt(1+x)*(1-3*x)^(3/2)*2*x)-1/(2*x)) \\ Altug Alkan, Nov 10 2015

Formula

D-finite with recurrence: -(n+1)*(n^2+n-3)*a(n) + 2*(n^3+3*n^2-4*n-3)*a(n-1) + 3*(n-1)*(n^2+3*n-1)*a(n-2) = 0. - R. J. Mathar, Feb 17 2016

From Mélika Tebni, Jan 24 2024: (Start)

a(n) = A005773(n+1) + A132894(n).

E.g.f.: (1+x)*exp(x)*(BesselI(0,2*x) + BesselI(1,2*x)). (End)

From Mélika Tebni, Jan 25 2024: (Start)

a(n) = Sum_{k=0..n} A189911(k)*binomial(n,k).

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