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

0,5

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

1/(1 + 1/(n + 1/(n^2 + 1/(n^3 + ...)))). Examples are given below. Cf. A005169 and A111317.

(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

Cf. A129182, A291874 (convolution inverse).

Cf. A005169, A111317, A224704.

Sequence in context: A073941 A005428 A355910 * A328262 A292800 A214041

Adjacent sequences: A143948 A143949 A143950 * A143952 A143953 A143954

Keyword

nonn

Author

Emeric Deutsch, Oct 09 2008

Extensions

b-file corrected and extended by Alois P. Heinz, Aug 24 2018

Status

approved