A115125 A sequence related to Catalan numbers A000108.

1, 2, 4, 16, 80, 448, 2688, 16896, 109824, 732160, 4978688, 34398208, 240787456, 1704034304, 12171673600, 87636049920, 635361361920, 4634400522240, 33985603829760, 250420238745600, 1853109766717440, 13765958267043840, 102618961627054080, 767411365211013120

Offset

0,2

Comments

Essentially identical to A025225.

The convolution of this sequence with the sequence {(-1)^n} is A064062 (see also A062992).

The sequence A064062 appears in the Derrida et al. 1992 reference (see A064094) for alpha=2, beta=1 (or alpha=1, beta=2).

Links

G. C. Greubel, Table of n, a(n) for n = 0..1000

B. Derrida, E. Domany and D. Mukamel, An exact solution of a one-dimensional asymmetric exclusion model with open boundaries, J. Stat. Phys. 69, 1992, 667-687; eqs. (20), (21), p. 672.

Formula

a(n) = C(n-1)*2^n, n>=1, a(0):=1, with C(n):=A000108(n) (Catalan).

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

a(n) = A025225(n), n>0. - R. J. Mathar, Aug 11 2008

G.f.: (3 - sqrt(1-8*x))/2 = 2 - U(0) where U(k)=1 - 2*x/U(k+1) ; (continued fraction, 1-step). - Sergei N. Gladkovskii, Oct 29 2012

G.f.: 2 - 1/Q(0), where Q(k)= 1 + (8*k+2)*x/(k+1 - x*(2*k+2)*(8*k+6)/(2*x*(8*k+6) + (2*k+3)/Q(k+1))); (continued fraction). - Sergei N. Gladkovskii, May 14 2013

Maple

a:= n-> `if`(n=0, 1, 2^n*binomial(2*n-2, n-1)/n):
seq(a(n), n=0..25);  # Alois P. Heinz, Jul 25 2022

Mathematica

a[0] = 1; a[n_] := 2^n*CatalanNumber[n - 1]; Table[a[n], {n, 0, 20}] (* Jean-François Alcover, Jul 09 2013 *)

Prog

(PARI) a(n)=if(n==0, 1, polcoeff((1-sqrt(1-8*x+x*O(x^n)))/2, n)); \\ Joerg Arndt, May 14 2013
(Magma) [1] cat [2^n*Binomial(2*n-2, n-1)/n: n in [1..30]]; // G. C. Greubel, May 03 2018

Crossrefs

Cf. A000108, A025225, A062992, A064062, A064094.

Sequence in context: A103619 A027436 A025225 * A326859 A213010 A000831

Adjacent sequences: A115122 A115123 A115124 * A115126 A115127 A115128

Keyword

nonn,easy,changed

Author

Wolfdieter Lang, Jan 13 2006

Status

approved