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A118093
Numbers of rooted hypermaps on the torus with n darts (darts are semi-edges in the particular case of ordinary maps).
6
1, 15, 165, 1611, 14805, 131307, 1138261, 9713835, 81968469, 685888171, 5702382933, 47168678571, 388580070741, 3190523226795, 26124382262613, 213415462218411, 1740019150443861, 14162920013474475, 115112250539595093, 934419385591442091, 7576722323539318101
OFFSET
3,2
LINKS
A. Mednykh and R. Nedela, Counting unrooted hypermaps on closed orientable surface, 18th Intern. Conf. Formal Power Series & Algebr. Comb., Jun 19, 2006, San Diego, California (USA).
A. Mednykh and R. Nedela, Enumeration of unrooted hypermaps of a given genus, Discr. Math., 310 (2010), 518-526. [From N. J. A. Sloane, Dec 19 2009]
Mednykh, A.; Nedela, R. Recent progress in enumeration of hypermaps, J. Math. Sci., New York 226, No. 5, 635-654 (2017) and Zap. Nauchn. Semin. POMI 446, 139-164 (2016), table 3
T. R. Walsh, Space-Efficient Generation of Nonisomorphic Maps and Hypermaps, J. Int. Seq. 18 (2015) # 15.4.3.
P. G. Zograf, Enumeration of Grothendieck's Dessins and KP Hierarchy, International Mathematics Research Notices, Volume 2015, Issue 24, 1 January 2015, 13533-13544.
FORMULA
Conjecture: +n*(5*n-17)*a(n) -15*(n-1)*(5*n-16)*a(n-1) +12*(20*n^2-103*n+140)*a(n-2) +32*(5*n-12)*(2*n-5)*a(n-3)=0. - R. J. Mathar, Apr 05 2018
G.f.: (1 - 7*x + 4*x^2 - (1 - 3*x)*sqrt(1 - 8*x))/(8*(1 + x)*(1 - 8*x)); equivalently, the g.f. can be rewritten as (y - 1)^3/(4*(y - 2)^2*(y + 1)), where y=G(2*x) with G the g.f. of A000108. - Gheorghe Coserea, Nov 06 2018
a(n) ~ 2^(3*n - 4) / 3 * (1 - 10/(3*sqrt(Pi*n))). - Vaclav Kotesovec, Nov 06 2018
MATHEMATICA
Table[Sum[2^k (4^(n - 2 - k) - 1) Binomial[n+k, k] / 3, {k, 0, n-3}], {n, 3, 25}] (* Vincenzo Librandi, Sep 16 2018 *)
PROG
(PARI) a(n) = sum(k=0, n-3, 2^k*(4^(n-2-k)-1)*binomial(n+k, k))/3; \\ Michel Marcus, Dec 11 2014
(PARI)
seq(N) = {
my(x='x+O('x^(N+2)), y=(1-sqrt(1-8*x))/(4*x));
Vec((y - 1)^3/(4*(y - 2)^2*(y + 1)));
};
seq(21) \\ Gheorghe Coserea, Nov 06 2018
(Magma) [&+[(2^k*(4^(n-2-k)-1)*Binomial(n+k, k))/3 : k in [0..n-3]]: n in [3..25]]; // Vincenzo Librandi, Sep 16 2018
CROSSREFS
KEYWORD
nonn
AUTHOR
Valery A. Liskovets, Apr 13 2006
EXTENSIONS
More terms from Michel Marcus, Dec 11 2014
STATUS
approved