A224747 Number of lattice paths from (0,0) to (n,0) that do not go below the x-axis and consist of steps U=(1,1), D=(1,-1) and H=(1,0), where H-steps are only allowed if y=1.

1, 0, 1, 1, 3, 5, 12, 23, 52, 105, 232, 480, 1049, 2199, 4777, 10092, 21845, 46377, 100159, 213328, 460023, 981976, 2115350, 4522529, 9735205, 20836827, 44829766, 96030613, 206526972, 442675064, 951759621, 2040962281, 4387156587, 9411145925, 20226421380

Offset

0,5

Comments

Apparently A125187 is even bisection. - R. J. Mathar, Jul 27 2013

Links

Alois P. Heinz, Table of n, a(n) for n = 0..1000

C. Banderier and M. Wallner, Lattice paths with catastrophes, SLC 77, Strobl - 12.09.2016, H(x).

Cyril Banderier and Michael Wallner, Lattice paths with catastrophes, arXiv:1707.01931 [math.CO], 2017.

Jean-Luc Baril and Sergey Kirgizov, Bijections from Dyck and Motzkin meanders with catastrophes to pattern avoiding Dyck paths, arXiv:2104.01186 [math.CO], 2021.

Jean-Luc Baril, Sergey Kirgizov, and Armen Petrossian, Dyck Paths with catastrophes modulo the positions of a given pattern, Australasian J. Comb. (2022) Vol. 84, No. 2, 398-418.

Formula

a(n) = Sum_{k=0..floor((n-2)/2)} A009766(2*n-3*k-3, k) for n >= 2. - Johannes W. Meijer, Jul 22 2013

HANKEL transform is A000012. HANKEL transform omitting a(0) is a period 4 sequence [0, -1, 0, 1, ...] = -A101455. - Michael Somos, Jan 14 2014

Given g.f. A(x), then 0 = A(x)^2 * (x^3 + 2*x^2 + x - 1) + A(x) * (-2*x^2 - 3*x + 2) + (2*x - 1). - Michael Somos, Jan 14 2014

0 = a(n)*(a(n+1) +2*a(n+2) +a(n+3) -a(n+4)) +a(n+1)*(2*a(n+1) +5*a(n+2) +a(n+3) -2*a(n+4)) +a(n+2)*(2*a(n+2) -a(n+3) -a(n+4)) +a(n+3)*(-a(n+3) +a(n+4)). - Michael Somos, Jan 14 2014

G.f.: (2 - 3*x - 2*x^2 + x * sqrt(1 - 4*x^2)) / (2 * (1 - x - 2*x^2 - x^3)). - Michael Somos, Jan 14 2014

a(2*n) = A125187(n).

D-finite with recurrence (-n+1)*a(n) +(n-1)*a(n-1) +6*(n-3)*a(n-2) +3*(-n+5)*a(n-3) +8*(-n+4)*a(n-4) +4*(-n+4)*a(n-5)=0. - R. J. Mathar, Sep 15 2021

Example

a(5) = 5: UHHHD, UDUHD, UUDHD, UHDUD, UHUDD.
a(6) = 12: UHHHHD, UDUHHD, UUDHHD, UHDUHD, UHUDHD, UHHDUD, UDUDUD, UUDDUD, UHHUDD, UDUUDD, UUDUDD, UUUDDD.
G.f. = 1 + x^2 + x^3 + 3*x^4 + 5*x^5 + 12*x^6 + 23*x^7 + 52*x^8 + 105*x^9 + ...

Maple

a:= proc(n) option remember; `if`(n<5, [1, 0, 1, 1, 3][n+1],
      a(n-1)+ (6*(n-3)*a(n-2) -3*(n-5)*a(n-3)
      -8*(n-4)*a(n-4) -4*(n-4)*a(n-5))/(n-1))
    end:
seq(a(n), n=0..40);

Mathematica

a[n_] := a[n] = If[n < 5, {1, 0, 1, 1, 3}[[n+1]], a[n-1] + (6*(n-3)*a[n-2] - 3*(n-5)*a[n-3] - 8*(n-4)*a[n-4] - 4*(n-4)*a[n-5])/(n-1)]; Table[a[n], {n, 0, 34}] (* Jean-François Alcover, Jun 20 2013, translated from Maple *)
a[ n_] := SeriesCoefficient[ (2 - 3 x - 2 x^2 + x Sqrt[1 - 4 x^2]) / (2 (1 - x - 2 x^2 - x^3)), {x, 0, n}] (* Michael Somos, Jan 14 2014 *)

Prog

(PARI) {a(n) = if( n<0, 0, polcoeff( (2 - 3*x - 2*x^2 + x * sqrt(1 - 4*x^2 + x * O(x^n)) ) / (2 * (1 - x - 2*x^2 - x^3)) n))} /* Michael Somos, Jan 14 2014 */

Crossrefs

Cf. A000108 (without H-steps), A001006 (unrestricted H-steps), A057977 (<=1 H-step).

Cf. A000012, A101455, A125187, A001405 (invert transform).

Sequence in context: A034758 A215109 A131322 * A036657 A047761 A349055

Adjacent sequences: A224744 A224745 A224746 * A224748 A224749 A224750

Keyword

nonn

Author

Alois P. Heinz, Apr 17 2013

Status

approved