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A318941 Number of Dyck paths with n nodes and altitude 2. 2
0, 0, 1, 4, 12, 35, 99, 274, 747, 2015, 5394, 14359, 38067, 100610, 265299, 698359, 1835922, 4821695, 12653739, 33188674, 87010587, 228039695, 597501714, 1565251879, 4099826787, 10737374210, 28118587299, 73630970599, 192799490322, 504817832015 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,4

LINKS

Colin Barker, Table of n, a(n) for n = 0..1000

Czabarka, É., Flórez, R., Junes, L., & Ramírez, J. L., Enumerations of peaks and valleys on non-decreasing Dyck paths, Discrete Mathematics (2018), 341(10), 2789-2807.

Index entries for linear recurrences with constant coefficients, signature (5,-7,2).

FORMULA

From Colin Barker, Apr 09 2019: (Start)

a(n) = 2^(-3-n)*(-3*4^n + 4*(3-sqrt(5))^n*(3+sqrt(5)) - 4*(-3+sqrt(5))*(3+sqrt(5))^n) for n>2.

a(n) = 5*a(n-1) - 7*a(n-2) + 2*a(n-3) n>5.

(End)

Note that Czabarka et al. give a g.f. for the whole triangle. - N. J. A. Sloane, Apr 09 2019

a(n) = A005248(n-1) -3*2^(n-3), n>=3. [Czabarka, Proposition 5 (2)] - R. J. Mathar, Apr 09 2019

MAPLE

(1-x)^2*x^2*(1+x)/(1-2*x)/(1-3*x+x^2) ;

taylor(%, x=0, 30) ;

gfun[seriestolist](%) ; # R. J. Mathar, Nov 25 2018

PROG

(PARI) concat([0, 0], Vec(x^2*(1 - x)^2*(1 + x) / ((1 - 2*x)*(1 - 3*x + x^2)) + O(x^40))) \\ Colin Barker, Apr 09 2019

CROSSREFS

A column of A318942.

Sequence in context: A246988 A340492 A084362 * A079736 A035045 A196859

Adjacent sequences:  A318938 A318939 A318940 * A318942 A318943 A318944

KEYWORD

nonn,easy

AUTHOR

N. J. A. Sloane, Sep 18 2018

STATUS

approved

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Last modified October 22 18:45 EDT 2021. Contains 348175 sequences. (Running on oeis4.)