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%I #52 Sep 08 2022 08:45:24
%S 1,15,165,1611,14805,131307,1138261,9713835,81968469,685888171,
%T 5702382933,47168678571,388580070741,3190523226795,26124382262613,
%U 213415462218411,1740019150443861,14162920013474475,115112250539595093,934419385591442091,7576722323539318101
%N Numbers of rooted hypermaps on the torus with n darts (darts are semi-edges in the particular case of ordinary maps).
%H Vincenzo Librandi, <a href="/A118093/b118093.txt">Table of n, a(n) for n = 3..500</a>
%H A. Mednykh and R. Nedela, <a href="http://garsia.math.yorku.ca/fpsac06/papers/9_ps_or_pdf.pdf">Counting unrooted hypermaps on closed orientable surface</a>, 18th Intern. Conf. Formal Power Series & Algebr. Comb., Jun 19, 2006, San Diego, California (USA).
%H A. Mednykh and R. Nedela, <a href="http://dx.doi.org/10.1016/j.disc.2009.03.033">Enumeration of unrooted hypermaps of a given genus</a>, Discr. Math., 310 (2010), 518-526. [From _N. J. A. Sloane_, Dec 19 2009]
%H Mednykh, A.; Nedela, R. <a href="https://doi.org/10.1007/s10958-017-3555-5">Recent progress in enumeration of hypermaps</a>, J. Math. Sci., New York 226, No. 5, 635-654 (2017) and Zap. Nauchn. Semin. POMI 446, 139-164 (2016), table 3
%H Timothy R. Walsh, <a href="http://www.info2.uqam.ca/~walsh_t/papers/GENERATING NONISOMORPHIC.pdf">Space-efficient generation of nonisomorphic maps and hypermaps</a>
%H T. R. Walsh, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL18/Walsh/walsh3.html">Space-Efficient Generation of Nonisomorphic Maps and Hypermaps</a>, J. Int. Seq. 18 (2015) # 15.4.3.
%H P. G. Zograf, <a href="https://doi.org/10.1093/imrn/rnv077">Enumeration of Grothendieck's Dessins and KP Hierarchy</a>, International Mathematics Research Notices, Volume 2015, Issue 24, 1 January 2015, 13533-13544.
%F 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
%F 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
%F a(n) ~ 2^(3*n - 4) / 3 * (1 - 10/(3*sqrt(Pi*n))). - _Vaclav Kotesovec_, Nov 06 2018
%t 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 *)
%o (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
%o (PARI)
%o seq(N) = {
%o my(x='x+O('x^(N+2)), y=(1-sqrt(1-8*x))/(4*x));
%o Vec((y - 1)^3/(4*(y - 2)^2*(y + 1)));
%o };
%o seq(21) \\ _Gheorghe Coserea_, Nov 06 2018
%o (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
%Y Cf. A000108, A000257, A118094, A214817, A214818.
%K nonn
%O 3,2
%A _Valery A. Liskovets_, Apr 13 2006
%E More terms from _Michel Marcus_, Dec 11 2014
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