A005161 -id:A005161 - cp's OEIS frontend

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

Paul Barry, Mar 03 2008

nonn

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

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.

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

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 *)

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

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)

A134565 Expansion of reversion of (x - 2*x^2) / (1 - x)^3.

Original entry on oeis.org

1, -1, 2, -3, 7, -12, 30, -55, 143, -273, 728, -1428, 3876, -7752, 21318, -43263, 120175, -246675, 690690, -1430715, 4032015, -8414640, 23841480, -50067108, 142498692, -300830572, 859515920, -1822766520, 5225264024, -11124755664, 31983672534, -68328754959

Offset: 1

Michael Somos, Nov 01 2007

sign

			G.f. = x - x^2 + 2*x^3 - 3*x^4 + 7*x^5 - 12*x^6 + 30*x^7 - 55*x^8 + 143*x^9 + ...

Paul Barry, Centered polygon numbers, heptagons and nonagons, and the Robbins numbers, arXiv:2104.01644 [math.CO], 2021.

Cf. A001764, A005161, A006013, A047749, A080956, A099576.

a[ n_] := With[ {m = Quotient[n, 2]}, If[n < 1, 0, -(-1)^n Binomial[n + m, n - m] / (2 m + 1)]]; (* Michael Somos, Oct 16 2015 *)
a[ n_] := If[n < 1, 0, SeriesCoefficient[ InverseSeries[ Series[(x - 2 x^2) / (1 - x)^3, {x, 0, n}]], {x, 0, n}]]; (* Michael Somos, Oct 16 2015 *)
a[n_] := (-1)^(n-1)*Binomial[2*n, n-1]*Hypergeometric2F1[-n+1, n, -2*n, -1] / n; Flatten[Table[a[n], {n, 1, 32}]] (* Detlef Meya, Dec 26 2023 *)

{a(n) = my( m = n\2); if( n<1, 0, -(-1)^n * binomial( n + m, n - m) / (2 * m + 1))};

{a(n) = if( n<1, 0, polcoeff( serreverse( (x - 2 * x^2) / (1 - x)^3 + x * O(x^n) ), n))};

{a(n) = if( n<1, 0, polcoeff( 1 / ( 1 + 1 / serreverse( x - x^3 + x * O(x^n) )), n))};

Given g.f. A(x), then 1 = (1/A(x) + 1/A(-x)) / 2.
a(n) = -(-1)^n * binomial(n + m, n - m) / (2*m + 1) where m = floor(n/2) if n>0.
From Michael Somos, Apr 13 2012 (Start)
a(n) = -(-1)^n * A047749(n) unless n=0. a(2*n) = - A001764(n) unless n=0. a(2*n + 1) = A006013(n).
Reversion of A080956 with offset 1.
Hankel transform is A005161 omitting first 1.
n * a(n) = -(-1)^n * A099576(n-1). (End)
D-finite with recurrence +8*n*(n+1)*a(n) -36*n*(n-2)*a(n-1) +6*(-9*n^2+18*n-14)*a(n-2) +27*(3*n-7)*(3*n-8)*a(n-3)=0. - R. J. Mathar, Sep 24 2021
a(n) = (-1)^(n-1)*binomial(2*n, n-1)*hypergeom([-n+1, n], [-2*n], -1) / n. - Detlef Meya, Dec 26 2023

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

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A134565 Expansion of reversion of (x - 2*x^2) / (1 - x)^3.

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