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A105695 Expansion of (1-x)*c(x/(1+x)), where c(x) is the g.f. of the Catalan numbers (A000108). 1
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 (list; graph; refs; listen; history; text; internal format)
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

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Last modified February 24 17:19 EST 2018. Contains 299624 sequences. (Running on oeis4.)