A318943 Number of Dyck paths with n nodes and altitude 3.

0, 0, 0, 1, 6, 21, 68, 208, 612, 1752, 4916, 13588, 37128, 100548, 270404, 723208, 1925844, 5110644, 13524872, 35713828, 94140900, 247806600, 651572660, 1711695508, 4493475336, 11789439876, 30917835908, 81053196808, 212426303892, 556607396532

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
• E. Czabarka et al, Enumerations of peaks and valleys on non-decreasing Dyck paths, Disc. Math. 341 (2018) 2789-2807.
• Index entries for linear recurrences with constant coefficients, signature (7,-17,16,-4).

Crossrefs

A column of A318942.

Programs

Maple

(1-x)^2*x^3*(1+x-3*x^2)/(1-2*x)^2/(1-3*x+x^2) ;
taylor(%,x=0,30) ;
gfun[seriestolist](%) ; # R. J. Mathar, Nov 25 2018

Mathematica

LinearRecurrence[{7, -17, 16, -4}, {0, 0, 0, 1, 6, 21, 68, 208}, 50] (* Paolo Xausa, May 24 2024 *)

PARI

concat([0,0,0], Vec(x^3*(1 - x)^2*(1 + x - 3*x^2) / ((1 - 2*x)^2*(1 - 3*x + x^2)) + O(x^40))) \\ Colin Barker, Apr 11 2019

Formula

a(n) = 8*A001906(n+1)-20*A001906(n)-2^(n-5)*(16+3*n), n>=4. - R. J. Mathar, Apr 09 2019

a(n) = 7*a(n-1) - 17*a(n-2) + 16*a(n-3) - 4*a(n-4) for n>7. - Colin Barker, Apr 11 2019