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

0,5

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, December 1, 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.

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)

Mathematica

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

Prog

(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

Crossrefs

Cf. A002026, A005043.

Sequence in context: A051163 A051450 A038508 * A244884 A002026 A026938

Adjacent sequences: A105692 A105693 A105694 * A105696 A105697 A105698

Keyword

easy,nonn

Author

Paul Barry, Apr 17 2005

Status

approved