A069721 Number of rooted unicursal planar maps with n edges and no vertices of valency 1 (unicursal means that exactly two vertices are of odd valency; there is an Eulerian path).

0, 0, 4, 40, 336, 2688, 21120, 164736, 1281280, 9957376, 77395968, 601968640, 4686094336, 36515020800, 284817162240, 2223764766720, 17379001958400, 135942415319040, 1064286014668800, 8338993950228480, 65388301768458240, 513094808135270400, 4028909667357818880

Offset

1,3

Links

Robert Israel, Table of n, a(n) for n = 1..1109

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

Youngja Park and SeungKyung Park, Enumeration of generalized lattice paths by string types, peaks, and ascents, Discrete Mathematics 339.11 (2016): 2652-2659.

Formula

a(n) = 2^(n-2)*(n-2)*binomial(2n-2, n-1)/n, n>1.

From Robert Israel, Nov 12 2016: (Start)

G.f.: 32*x^3/(sqrt(1-8*x)*(1+sqrt(1-8*x))^3).

E.g.f.: ((1-6*x)/4)*exp(4*x)*I_0(4*x)+(3/2)*exp(4*x)*I_1(4*x)+x/2-1/4, where I_0 and I_1 are modified Bessel functions of the first kind.

a(n+1) = (4*(n-1)*(2*n-1)/((n+1)*(n-2)))*a(n).

a(n) ~ 8^n/(16*sqrt(Pi*n)). (End)

From Amiram Eldar, Jan 16 2024: (Start)

Sum_{n>=3} 1/a(n) = 11/14 - 26*arcsin(1/(2*sqrt(2)))/(7*sqrt(7)).

Sum_{n>=3} (-1)^(n+1)/a(n) = 37*log(2)/27 - 13/18. (End)

Example

G.f. = 4*x^3 + 40*x^4 + 336*x^5 + 2688*x^6 + 21120*x^7 + 164736*x^8 + ...

Maple

0, seq(2^(n-2)*(n-2)*binomial(2*n-2, n-1)/n, n=2..30); # Robert Israel, Nov 12 2016

Mathematica

a[ n_] := SeriesCoefficient[ ((1 - Sqrt[1 - 8 x])/2)^3 / (2 Sqrt[1 - 8 x] ), {x, 0, n}]; (* Michael Somos, Nov 13 2016 *)

Prog

(Magma) [0] cat [2^(n-2)*(n-2)*Binomial(2*n-2, n-1)/n: n in [2..25]]; // Vincenzo Librandi, Nov 13 2016

Crossrefs

Cf. A069720, A069722, A069723.

Sequence in context: A043031 A121126 A145730 * A223176 A279574 A327675

Adjacent sequences: A069718 A069719 A069720 * A069722 A069723 A069724

Keyword

easy,nonn

Author

Valery A. Liskovets, Apr 07 2002

Status

approved