A105695 Expansion of (1-x)*c(x/(1+x)), where c(x) is the g.f. of the Catalan numbers (A000108).

1, 0, 0, 1, 2, 5, 12, 30, 76, 196, 512, 1353, 3610, 9713, 26324, 71799, 196938, 542895, 1503312, 4179603, 11662902, 32652735, 91695540, 258215664, 728997192, 2062967382, 5850674704, 16626415975, 47337954326

Offset: 0

Author

Paul Barry, Apr 17 2005

Keywords

• easy
• nonn

Comments

Apply the Riordan array (1-x,x/(1+x)) to C(n)=A000108(n).

Alternatively, this sequence corresponds to the number of positive walks with n steps {-1,0,1} starting at the origin, ending at altitude 2, and staying strictly above the x-axis. - D. Nguyen, Dec 01 2016.

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..200
• C. Banderier, C. Krattenthaler, A. Krinik, D. Kruchinin, V. Kruchinin, D. Nguyen, and M. Wallner, Explicit formulas for enumeration of lattice paths: basketball and the kernel method, arXiv:1609.06473 [math.CO], 2016.

Crossrefs

Cf. A002026, A005043.

Programs

Mathematica

CoefficientList[Series[(1-x^2)*(1-Sqrt[(1-3*x)/(1+x)])/(2*x), {x, 0, 20}], x] (* Vaclav Kotesovec, Feb 01 2014 *)

PARI

x='x+O('x^50); Vec((1-x^2)*(1-sqrt((1-3*x)/(1+x)))/(2*x)) \\ G. C. Greubel, Mar 16 2017

Formula

G.f.: (1-x^2)*(1-sqrt((1-3*x)/(1+x)))/(2*x).

Let b(n) = Sum_{k=0..n} (-1)^(n-k)*binomial(n, k)*C(k) = A005043(n); then a(n) = b(n) - b(n-2).

Conjecture: (n+1)*a(n)+(2-3n)*a(n-1) +(1-n)*a(n-2)+3*(n-4)*a(n-3)=0. - R. J. Mathar, Dec 13 2011

a(n) ~ 3^(n-1/2) / (sqrt(Pi) * n^(3/2)). - Vaclav Kotesovec, Feb 01 2014

From Peter Bala, Oct 29 2015: (Start)

a(n) = Sum_{k = 1..floor((n-1)/2)} binomial(n-2,2*k-1)*Catalan(k) for n >= 1.

(n+1)*(n-3)*a(n) = (n-2)*(2*n-3)*a(n-1) + 3*(n-2)*(n-3)*a(n-2) with a(2) = 0, a(3) = 1. Mathar's 4-term recurrence above follows easily from this. (End)