login
A115125
A sequence related to Catalan numbers A000108.
2
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
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
KEYWORD
nonn,easy
AUTHOR
Wolfdieter Lang, Jan 13 2006
STATUS
approved