A069731 - OEIS

%I #21 May 01 2024 11:29:04

%S 1,5,28,168,1056,6864,45760,311168,2149888,15049216,106502144,

%T 760729600,5477253120,39710085120,289650032640,2124100239360,

%U 15651264921600,115819360419840,860372391690240

%N 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).

%H V. A. Liskovets and T. R. S. Walsh, <a href="http://dx.doi.org/10.1016/j.disc.2003.09.015">Enumeration of Eulerian and unicursal planar maps</a>, Discr. Math., 282 (2004), 209-221.

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

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

%F D-finite with recurrence: 4*(2*n+1)*a(n-1) - (n+2)*a(n) = 0, a(1) = 1. - _Georg Fischer_, May 23 2021

%F From _Peter Bala_, Apr 29 2024: (Start)

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

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

%F 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)

%p 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

%t Table[2^(n-2) CatalanNumber[n+1], {n, 1, 19}] (* _Jean-François Alcover_, Aug 28 2019 *)

%Y Cf. A069724, A069720, A000108, A003645.

%Y Cf. A000346, A001006, A136576.

%K easy,nonn

%O 1,2

%A _Valery A. Liskovets_, Apr 07 2002