OFFSET
0,5
COMMENTS
Column sums of A129182.
LINKS
Alois P. Heinz, Table of n, a(n) for n = 0..2000 (first 1001 terms from Vincenzo Librandi)
Paul Barry, On sequences with {-1, 0, 1} Hankel transforms, arXiv preprint arXiv:1205.2565 [math.CO], 2012. - From N. J. A. Sloane, Oct 18 2012
FORMULA
G.f.: 1/(1 - x/(1 - x^3/(1 - x^5/(1 - x^7/(1 - x^9/(1 - ...
Derivation: the g.f. G(x,z) of Dyck paths, where x marks area and z marks semilength, satisfies G(x,z)=1+x*z*G(x,z)*G(x,x^2*z). Set z=1.
From Peter Bala, Dec 26 2012: (Start)
Let F(x) denote the o.g.f. of this sequence. For positive integer n >= 3, the real number F(1/n) has the simple continued fraction expansion 1 + 1/(n-2 + 1/(1 + 1/(n^2-2 + 1/(1 + 1/(n^3-2 + 1/(1 + ...)))))).
For n >= 1, F(-1/n) has the simple continued fraction expansion
(End)
G.f.: A(x) = 1/(1 - x/(1-x + x/(1+x^2 + x^4/(1-x^3 - x^2/(1+x^4 - x^7/(1-x^5 + x^3/(1+x^6 + x^10/(1-x^7 - x^4/(1+x^8 - x^13/(1-x^9 + x^5/(1+x^10 + x^16/(1 + ...)))))))))))), a continued fraction. - Paul D. Hanna, Aug 08 2016
a(n) ~ c / r^n, where r = 0.66290148514884371255690407749133031115536799774051... and c = 0.337761150388539773466092171229604432776662930886727976914... . - Vaclav Kotesovec, Feb 17 2017, corrected Nov 04 2021
From Peter Bala, Jul 04 2019: (Start)
O.g.f. as a ratio of q-series: N(q)/D(q), where N(q) = Sum_{n >= 0} (-1)^n*q^(2*n^2+n)/( (1-q^2)*(1-q^4)*...*(1-q^(2*n)) ) and D(q) = Sum_{n >= 0} (-1)^n*q^(2*n^2-n)/( (1-q^2)*(1-q^4)*...*(1-q^(2*n)) ). Cf. A224704.
D(q) has its least positive (and simple) real zero at x = 0.66290 14851 48843 71255 69040 ....
a(n) ~ c*d^n, where d = 1/x = 1.5085197761707628638804960 ... and c = - N(x)/(x*D'(x)) = 0.3377611503885397734660921 ... (the prime indicates differentiation w.r.t. q). (End)
EXAMPLE
a(5)=3 because we have UDUUDD, UUDDUD and UDUDUDUDUD, where U=(1,1) and D=(1,-1).
From Peter Bala, Dec 26 2012: (Start)
F(1/10) = sum {n >= 0} a(n)/10^n has the simple continued fraction expansion 1 + 1/(8 + 1/(1 + 1/(98 + 1/(1 + 1/(998 + 1/(1 + ...)))))).
F(-1/10) = sum {n >= 0} (-1)^n*a(n)/10^n has the simple continued fraction expansion 1/(1 + 1/(10 + 1/(100 + 1/(1000 + ...)))).
(End)
MAPLE
g:=1/(1-x/(1-x^3/(1-x^5/(1-x^7/(1-x^9/(1-x^11/(1-x^13/(1-x^15)))))))): gser:= series(g, x=0, 45): seq(coeff(gser, x, n), n=0..44);
# second Maple program:
b:= proc(x, y, k) option remember;
`if`(y<0 or y>x or k<0 or k>x^2/2-(y-x)^2/4, 0,
`if`(x=0, 1, b(x-1, y-1, k-y+1/2) +b(x-1, y+1, k-y-1/2)))
end:
a:= n-> add(b(2*n-4*t, 0, n), t=0..n/2):
seq(a(n), n=0..50); # Alois P. Heinz, Aug 24 2018
MATHEMATICA
terms = 50; CoefficientList[1/(1+ContinuedFractionK[-x^(2i-1), 1, {i, 1, Sqrt[terms]//Ceiling}]) + O[x]^terms, x] (* Jean-François Alcover, Jul 11 2018 *)
PROG
(PARI) N=66; q = 'q +O('q^N);
G(k) = if(k>N, 1, 1 - q^(k+1) / G(k+2) );
gf = 1 / G(0);
Vec(gf) \\ Joerg Arndt, Jul 06 2013
CROSSREFS
KEYWORD
nonn
AUTHOR
Emeric Deutsch, Oct 09 2008
EXTENSIONS
b-file corrected and extended by Alois P. Heinz, Aug 24 2018
STATUS
approved