A138164 Row sums of Riordan array (c(-x^2),xc(-x^2)^2)^(-1) where c(x) is the g.f. of A000108.

1, 1, 2, 4, 9, 20, 47, 109, 262, 622, 1516, 3653, 8988, 21883, 54213, 133004, 331233, 817432, 2044151, 5068346, 12716872, 31651555, 79636493, 198843284, 501466519, 1255489165, 3172569392, 7961388439, 20152910577, 50674576772

Offset: 0

Author

Paul Barry, Mar 03 2008

Keywords

• nonn

Comments

Hankel transform is A005161.

A transform of the F(n+1) by (1,x(1-x^2))^(-1).

(c(-x^2),xc(-x^2)^2)^(-1) factors as (1,x(1-x^2))^(-1)*(1/(1-x^2),x/(1-x^2)).

It appears that a(n) is the number of Dyck paths (A000108) of semilength n in which all non-return descents are of even length (a return descent is a maximal sequence of downsteps that returns the path to ground level). For example, a(4) = 9 counts, among others, UUUDDUDD and UDUUUDDD but not UUDUUDDD. - David Callan, Nov 13 2021

Links

• Vaclav Kotesovec, Table of n, a(n) for n = 0..2000
• J.-L. Baril, R. Genestier, A. Giorgetti, and A. Petrossian, Rooted planar maps modulo some patternss, Preprint 2016.

Programs

Maple

vx := 2/sqrt(3)*sin(arcsin(x*3*sqrt(3)/2)/3) ;
A138164 := proc(n)
        1/(1-vx-vx^2) ;
        coeftayl(%,x=0,n) ;
        subs(4^(1/2)=2,%) ;
end proc: # R. J. Mathar, Jul 28 2016

Mathematica

CoefficientList[Series[1/(1/3 - 2*Sin[1/3*ArcSin[3*Sqrt[3]*x/2]]/Sqrt[3] + 2*Cos[2/3*ArcSin[3*Sqrt[3]*x/2]]/3), {x, 0, 30}], x] (* Vaclav Kotesovec, Nov 15 2021 *)

PARI

my(x='x+O('x^66)); v=serreverse(x*(1-x^2)); Vec(1/(1-v-v^2)) \\ Joerg Arndt, Feb 24 2015

Formula

G.f.: 1/(1-v-v^2) where v=(2/sqrt(3))*sin(arcsin(x*3*sqrt(3)/2)/3) is the series reversion of x(1-x^2). [Corrected by Paul D. Hanna, Feb 24 2015]

From Gary W. Adamson, Jul 07 2011: (Start)

Let M = the production matrix:

1, 1, 0, 0, 0, 0, 0, 0, ...

1, 0, 1, 0, 0, 0, 0, 0, ...

1, 1, 0, 1, 0, 0, 0, 0, ...

1, 0, 1, 0, 1, 0, 0, 0, ...

1, 1, 0, 1, 0, 1, 0, 0, ...

1, 0, 1, 0, 1, 0, 1, 0, ...

1, 1, 0, 1, 0, 1, 0, 1, ...

...

a(n) = top left term of M^n. a(n+1) = sum of top row terms of M^n. Example: top row of M^3 = (4, 3, 1, 1), where a(3) = 4 and a(4) = 9 = (4 + 3 + 1 + 1). (End)

v(x) = Sum_{n>=1} A001764(n-1)*x^(2*n-1). - Paul D. Hanna, Feb 24 2015

Conjecture: -8*n*(n-1)*a(n) + 12*(n+3)*(n-1)*a(n-1) + 2*(89*n^2-512*n+651)*a(n-2) + 3*(-127*n^2+715*n-966)*a(n-3) + 3*(-247*n^2+2431*n-5698)*a(n-4) + 9*(75*n-341)*(3*n-16)*a(n-5) - 72*(3*n-14)*(3*n-16)*a(n-6) = 0. - R. J. Mathar, Feb 24 2015

From Vaclav Kotesovec, Nov 15 2021: (Start)

Recurrence (of order 4): 4*(n-1)*n*(11*n^2 - 50*n + 48)*a(n) = 12*(n-1)*(11*n^3 - 50*n^2 + 66*n - 40)*a(n-1) + (253*n^4 - 2294*n^3 + 6379*n^2 - 5898*n + 720)*a(n-2) - 3*(297*n^4 - 2538*n^3 + 7347*n^2 - 7946*n + 2000)*a(n-3) + 3*(3*n - 10)*(3*n - 8)*(11*n^2 - 28*n + 9)*a(n-4).

a(n) ~ (45*(1 - (-1)^n) + 26*sqrt(3)*(1 + (-1)^n)) * 3^(3*n/2) / (sqrt(Pi) * n^(3/2) * 2^(n + 1/2)). (End)