A322928 - OEIS

%I #31 Mar 02 2023 09:09:17

%S 1,9,118,1773,28650,484578,8457708,151054173,2745685954,50606020854,

%T 943283037684,17746990547634,336517405188900,6423775409047716,

%U 123332141503711704,2379824766494404317,46124764901514110898,897483137740689843054,17524230350476917414180

%N a(0)=1; for n>0, a(n) is the number of rooted 3-regular maps with 2n vertices on the projective plane.

%H Valentin Bonzom, Guillaume Chapuy, Maciej Dolega, <a href="https://doi.org/10.5802/alco.268">Enumeration of non-oriented maps via integrability</a>, Alg. Combin. 5 (6) (2022) p 1363-1390, A.3.

%H Evgeniy Krasko and Alexander Omelchenko, <a href="https://doi.org/10.1016/j.disc.2018.07.013">Enumeration of r-regular maps on the torus. Part I: Rooted maps on the torus, the projective plane and the Klein bottle. Sensed maps on the torus</a>, Discrete Mathematics (2019) Vol. 342, Issue 2, 584-599. Also <a href="https://arxiv.org/abs/1709.03225">arXiv:1709.03225 [math.CO]</a>. See Th. 3.3 and Table 2.

%F Theorem 3.3 gives an explicit formula.

%F From _Vaclav Kotesovec_, Dec 30 2022: (Start)

%F Recurrence: (n-3)*(n-2)*(n-1)*n*(n+1)*a(n) = 12*(n-3)*(n-2)*(n-1)*(108*n^2 - 432*n + 505)*a(n-2) - 576*(n-3)*(3*n - 10)*(3*n - 8)*(108*n^2 - 648*n + 1049)*a(n-4) + 995328*(n-4)*(3*n - 16)*(3*n - 14)*(3*n - 10)*(3*n - 8)*a(n-6).

%F a(n) ~ Gamma(1/4) * 2^(2*n - 5/4) * 3^(3*n/2 + 5/4)  / (Pi * n^(5/4)) * (1 - 2^(7/4)*sqrt(Pi)/(Gamma(1/4)*3^(3/4)*n^(1/4)) + Pi/(Gamma(1/4)^2*sqrt(3*n))).

%F (End)

%p A[0]:= 1: A[1]:= 9: A[2]:= 118: A[3]:= 1773: A[4]:= 28650: A[5]:= 484578:

%p for n from 6 to 20 do

%p   A[n]:= 995328*(n - 4)*(3*n - 16)*(3*n - 14)*(3*n - 10)*(3*n - 8)*A[n - 6]/((n - 3)*(n - 2)*(n - 1)*n*(n + 1)) - 576*(3*n - 10)*(3*n - 8)*(108*n^2 - 648*n + 1049)*A[n - 4]/((n - 2)*(n - 1)*n*(n + 1)) + 12*(108*n^2 - 432*n + 505)*A[n - 2]/(n*(n + 1))

%p od:

%p seq(A[i],i=0..20); # _Robert Israel_, Dec 30 2022

%t a[n_] := -((2^(2 n + 1) (3 n)!!)/((n + 1)! n!!)) + (3 2^(2 n))/(n + 1)!! Sum[(3^k (2 k - 1)!! (3 n - 2 k - 1)!!)/(2^k k! (n - k)!), {k, 0, n}];

%t Table[a[n], {n, 0, 20}] (* _Andrey Zabolotskiy_, Dec 29 2022 *)

%Y Cf. A002005 (genus 0)

%K nonn

%O 0,2

%A _Evgeniy Krasko_, Dec 31 2018

%E Added initial term a(0)=1 to match Taylor series expansion in Theorem 3.3. - _N. J. A. Sloane_, Jan 11 2019

%E Terms a(11) and beyond from _Andrey Zabolotskiy_, Dec 29 2022