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

0,5

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).

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

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 *)

Prog

(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

Crossrefs

A column of A318942.

Sequence in context: A107653 A123653 A364636 * A200761 A169687 A302448

Adjacent sequences: A318940 A318941 A318942 * A318944 A318945 A318946

Keyword

nonn,easy,changed

Author

N. J. A. Sloane, Sep 18 2018

Status

approved