A143951 Number of Dyck paths such that the area between the x-axis and the path is n.
1, 1, 1, 1, 2, 3, 4, 6, 9, 14, 21, 31, 47, 71, 107, 161, 243, 367, 553, 834, 1258, 1898, 2863, 4318, 6514, 9827, 14824, 22361, 33732, 50886, 76762, 115796, 174680, 263509, 397508, 599647, 904579, 1364576, 2058489, 3105269, 4684359, 7066449, 10659877, 16080632, 24257950, 36593598, 55202165, 83273553, 125619799, 189499952
Offset: 0
Keywords
Examples
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)
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
Programs
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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 *) -
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
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)
Extensions
b-file corrected and extended by Alois P. Heinz, Aug 24 2018
Comments