A318944 Number of Dyck paths with n nodes and altitude 4.

0, 0, 0, 0, 1, 8, 32, 114, 376, 1177, 3549, 10406, 29861, 84249, 234502, 645625, 1761765, 4772534, 12851261, 34434561, 91890118, 244385617, 648139821, 1714976054, 4529163125, 11942440233, 31448759302, 82727323369, 217426319541, 571033273142

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 (9,-31,50,-36,8).

Crossrefs

A column of A318942.

Programs

Maple

(1-x)^2*x^4*(1+x-8*x^2+7*x^3)/(1-2*x)^3/(1-3*x+x^2) ;
taylor(%,x=0,30) ;
gfun[seriestolist](%) ; # R. J. Mathar, Nov 25 2018

Mathematica

LinearRecurrence[{9,-31,50,-36,8},{0,0,0,0,1,8,32,114,376,1177},30] (* Harvey P. Dale, Nov 03 2019 *)

PARI

concat([0,0,0,0], Vec(x^4*(1 - x)^2*(1 + x - 8*x^2 + 7*x^3) / ((1 - 2*x)^3*(1 - 3*x + x^2)) + O(x^40))) \\ Colin Barker, Apr 11 2019

Formula

a(n) = 9*a(n-1) - 31*a(n-2) + 50*a(n-3) - 36*a(n-4) + 8*a(n-5) for n>9. - Colin Barker, Apr 11 2019