A118093 Numbers of rooted hypermaps on the torus with n darts (darts are semi-edges in the particular case of ordinary maps).

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

Vincenzo Librandi, Table of n, a(n) for n = 3..500

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

Timothy R. Walsh, Space-efficient generation of nonisomorphic maps and hypermaps

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

Cf. A000108, A000257, A118094, A214817, A214818.

Sequence in context: A164599 A016291 A229406 * A167615 A210326 A016234

Adjacent sequences: A118090 A118091 A118092 * A118094 A118095 A118096

Keyword

nonn

Author

Valery A. Liskovets, Apr 13 2006

Extensions

More terms from Michel Marcus, Dec 11 2014

Status

approved