A063016 - OEIS

A063016

a(n) is the product of Catalan(n) and (2^(n+1) - 1).

1, 3, 14, 75, 434, 2646, 16764, 109395, 730730, 4973826, 34381412, 240728670, 1703826292, 12170930700, 87633375480, 635351667075, 4634365164570, 33985474184970, 250419761106900, 1853107999454250, 13765951702923420, 102618937160787060, 767411273728449480

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OFFSET

0,2

COMMENTS

From Michael Wallner, Nov 03 2021: (Start)

a(n) is also the number of n X 2 Young tableaux with (possibly) vertical walls. The entries in cells that are separated by such a wall do not have to obey any order constraints. See Banderier, Wallner 2021 and Banderier et al. 2018.

a(n) is also the number of binary trees with n vertices and marked leaves, where at least 1 leaf has to be marked. Banderier, Wallner 2021 give a bijection to n X 2 Young tableaux with vertical walls. (End)

LINKS

Harry J. Smith, Table of n, a(n) for n = 0..200

C. Banderier, P. Marchal, and M. Wallner, Rectangular Young tableaux with local decreases and the density method for uniform random generation, GASCOM 2018. Vol. 2113. CEUR Workshop Proceedings, 2018.

C. Banderier and M. Wallner, Young tableaux with periodic walls: counting with the density method Sém. Lothar. Combin. 85B (2021), Art. 47, 12 pp.

FORMULA

a(n) = Catalan(n)*(2^(n+1) - 1) = A000108(n) * A126646(n).

D-finite with recurrence: a(n) = 2*(2*n-1)*(3*n*a(n-1)-4*(2*n-3)*a(n-2))/((n+1)*n). - Georg Fischer, Jun 06 2021

G.f.: A(x) = (sqrt(1-4*x) - sqrt(1-8*x))/(2*x).

G.f.: G(0)/(2*x) where G(k) = 1 - 2^k/(1 - 2*x*(2*k-1)/(2*x*(2*k-1) - 2^k*(k+1)/G(k+1))); (continued fraction, 3rd kind, 3-step). - Sergei N. Gladkovskii, Jul 24 2012

From Peter Bala, Aug 17 2021: (Start)

a(n) = Sum_{k = 0..n} A046521(n,k)*Catalan(k).

G.f.: A(x) = 1/sqrt(1 - 4*x)*c(x/(1 - 4*x)), where c(x) = (1 - sqrt(1 - 4*x))/(2*x) is the g.f. of the Catalan numbers A000108. Inversely, c(x) = 1/sqrt(1 + 4*x)*A(x/(1 + 4*x)).

Series reversion of x*A(x) = x*(1 - 3*x + 4*x^2*c(-2*x^2)). (End)

MATHEMATICA

Table[CatalanNumber[n]*(2^(n+1)-1), {n, 0, 20}] (* Harvey P. Dale, Oct 20 2014 *)

PROG

(PARI) a(n) = (2^(n + 1) - 1)*binomial(2*n, n)/(n + 1); \\ Harry J. Smith, Aug 16 2009

(Sage)

def A063016(n) :

return (8^(n+1)-4^(n+1))*factorial(n-1/2)/(4*sqrt(pi)*factorial(n+1))

[A063016(i) for i in (0..20)] # Peter Luschny, Jul 24 2012

CROSSREFS

Cf. A000108, A063017, A126646, A007317, A046521.

Sequence in context: A303034 A026004 A200718 * A246455 A133798 A100937