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
Examples
G.f. = 4*x^3 + 40*x^4 + 336*x^5 + 2688*x^6 + 21120*x^7 + 164736*x^8 + ...
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.
Programs
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Magma
[0] cat [2^(n-2)*(n-2)*Binomial(2*n-2, n-1)/n: n in [2..25]]; // Vincenzo Librandi, Nov 13 2016
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Maple
0, seq(2^(n-2)*(n-2)*binomial(2*n-2, n-1)/n, n=2..30); # Robert Israel, Nov 12 2016
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Mathematica
a[ n_] := SeriesCoefficient[ ((1 - Sqrt[1 - 8 x])/2)^3 / (2 Sqrt[1 - 8 x] ), {x, 0, n}]; (* Michael Somos, Nov 13 2016 *)
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)