A069731 Number of unicursal planar maps with n edges rooted at a vertex of odd valency (unicursal means that exactly two vertices are of odd valency; there is an Eulerian path).

1, 5, 28, 168, 1056, 6864, 45760, 311168, 2149888, 15049216, 106502144, 760729600, 5477253120, 39710085120, 289650032640, 2124100239360, 15651264921600, 115819360419840, 860372391690240

Offset

1,2

Links

Table of n, a(n) for n=1..19.

V. A. Liskovets and T. R. S. Walsh, Enumeration of Eulerian and unicursal planar maps, Discr. Math., 282 (2004), 209-221.

Formula

a(n) = 2^(n-2)*C_(n+1), where C_n stands for the Catalan numbers (A000108).

a(n) = A003645(n+2)/4.

D-finite with recurrence: 4*(2*n+1)*a(n-1) - (n+2)*a(n) = 0, a(1) = 1. - Georg Fischer, May 23 2021

From Peter Bala, Apr 29 2024: (Start)

a(n) = Sum_{k = 0..n} binomial(n, 2*k)*Catalan(k)*4^(n-k-1).

O.g.f.: A(x) = (1 - 4*x - 8*x^2 - sqrt(1 - 8*x))/(32*x^2).

A(x) = series reversion of x*c(-x)/(1 + 4*x), where c(x) = (1 - sqrt(1 - 4*x))/(2*x) is the g.f. of the Catalan numbers A000108 and c(-x)/(1 + 4*x) is the g.f. of (-1)^n*A000346(n). (End)

Maple

Z:=-(1-4*z-sqrt(1-4*z))/sqrt(1-4*z)/64: Zser:=series(Z, z=0, 32): seq(coeff(Zser*2^(n+1), z, n), n=3..24); # Zerinvary Lajos, Jan 01 2007

Mathematica

Table[2^(n-2) CatalanNumber[n+1], {n, 1, 19}] (* Jean-François Alcover, Aug 28 2019 *)

Crossrefs

Cf. A069724, A069720, A000108, A003645.

Cf. A000346, A001006, A136576.

Sequence in context: A371778 A083316 A027284 * A272046 A370025 A292871

Adjacent sequences: A069728 A069729 A069730 * A069732 A069733 A069734

Keyword

easy,nonn,changed

Author

Valery A. Liskovets, Apr 07 2002

Status

approved