A140662 Number of possible column states for self-avoiding polygons in a slit of width n.

1, 3, 8, 20, 50, 126, 322, 834, 2187, 5797, 15510, 41834, 113633, 310571, 853466, 2356778, 6536381, 18199283, 50852018, 142547558, 400763222, 1129760414, 3192727796, 9043402500, 25669818475, 73007772801, 208023278208, 593742784828, 1697385471210, 4859761676390

Offset: 1

Author

R. J. Mathar, Jul 11 2008

Keywords

• easy
• nonn

Comments

Number of Dyck (n+1)-paths whose maximum ascent length is 2. - David Scambler, Aug 22 2012

Number of (n+1)-Motzkin-paths with at least one up-step (see A001006 and the Python program). - Peter Luschny, Dec 03 2024

Examples

			The 20 Motzkin-paths of length 5 with at least one up-step are: UUDDF, UUDFD, UUFDD, UDUDF, UDUFD, UDFUD, UDFFF, UFUDD, UFDUD, UFDFF, UFFDF, UFFFD, FUUDD, FUDUD, FUDFF, FUFDF, FUFFD, FFUDF, FFUFD, FFFUD.
		

Links

• J. Alvarez, E.J. Janse van Rensburg et al. Self-avoiding walks and polygons in slits.
• Xiaomei Chen, Counting humps and peaks in Motzkin paths with height k, arXiv:2412.00668 [math.CO], 2024. See p. 7.
• Louis Marin, Counting Polyominoes in a Rectangle b X h, arXiv:2406.16413 [cs.DM], 2024. See p. 148.

Crossrefs

Cf. A055151, A080159, A088617, A107131, A097610.

Cf. A001006.

Column k=2 of A203717 (shifted).

Programs

Maple

a := n -> n*(n + 1)*hypergeom([1, -n/2 + 1, 1/2 - n/2], [2, 3], 4)/2:
seq(simplify(a(n)), n = 1..30);  # Peter Luschny, Dec 03 2024

Python

# A generator of the Motzkin-paths with at least one up-step.
C = str.count
def aGen(n: int): # -> Generator[str, Any, list[str]]
    a = [""]
    for w in a:
        if len(w) == n + 1:
            if (C(w, "U") > 0 and C(w, "U") == C(w, "D")): yield w
        else:
            for j in "UDF":
                u = w + j
                if C(u, "U") >= C(u, "D"): a += [u]
    return a
for n in range(1, 6):
    SAP = [w for w in aGen(n)]
    print(len(SAP), ":", SAP)  # Peter Luschny, Dec 03 2024

Formula

a(n) = Sum_{m=1..[(n+1)/2]} (n+1)!/((n+1-2m)!m!(m+1)!).

a(n) = A001006(n + 1) - 1. [Corrected by Peter Luschny, Dec 03 2024]

D-finite with recurrence (n+3)*a(n) + (-4*n-7)*a(n-1) + (2*n+3)*a(n-2) + (4*n-5)*a(n-3) + 3*(-n+2)*a(n-4) = 0. - R. J. Mathar, Nov 01 2021

From Peter Luschny, Dec 03 2024: (Start)

a(n) = (1/2)*n*(n + 1)*hypergeom([1, -n/2 + 1, 1/2 - n/2], [2, 3], 4).

a(n) = n!*[x^n]((exp(x)*(-x^3 + 2*(2*x - 3)*x*BesselI(0,2*x) + (x*(5*x - 4) + 6)*BesselI(1, 2* x)))/x^3). (End)