A318941 Number of Dyck paths with n nodes and altitude 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

Offset: 0

Author

N. J. A. Sloane, Sep 18 2018

Keywords

• nonn
• easy

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).

Crossrefs

A column of A318942.

Programs

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

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

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