A035319 Number of rooted maps of genus n with one vertex and one face; the maps are considered on orientable surfaces and contain 2n edges.

1, 1, 21, 1485, 225225, 59520825, 24325703325, 14230536445125, 11288163762500625, 11665426077721040625, 15230046989184655753125, 24515740420894935215128125, 47702727710977364941596305625

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn

Comments

a(n) is also the number of 2-permutations in Sym(4n-1), for n>1 (see Doignon and Labarre). - Anthony Labarre, Jun 19 2007

Links

• Gheorghe Coserea, Table of n, a(n) for n = 0..200
• Nikita Alexeev and Peter Zograf, Hultman numbers, polygon gluings and matrix integrals, arXiv preprint arXiv:1111.3061 [math.PR], 2011.
• J.-P. Doignon and A. Labarre, On Hultman Numbers, J. Integer Seq., 10 (2007), 13 pages.
• T. R. S. Walsh and A. B. Lehman, Counting rooted maps by genus. I, J. Comb. Theory B 13 (1972), 192-218 (Tab.1).

Crossrefs

Right-hand diagonal of A035309.

Cf. A035309.

Programs

Maple

A035319 := proc(n)
    (4*n)!/4^n/(2*n+1)! ;
end proc:
seq(A035319(n),n=0..10) ; # R. J. Mathar, Jun 12 2018

PARI

a(n) = (4*n)!/((2*n+1)!*4^n);  \\ Gheorghe Coserea, Jan 21 2017

Formula

a(n) = A035318(2*n). - Valery A. Liskovets, Apr 13 2006

It appears that this is given by the formula (4n)!/2^{2n}(2n+1)! = (4n-1)!!/(2n+1). (This sequence arose -- conjecturally, but it shouldn't be too hard to make it rigorous -- as the unique nontrivial Betti number of a certain poset associated to the hyperoctahedral group.) - Eric M. Rains (rains(AT)caltech.edu), Jan 24 2006

a(n) = (4n)!/(2^(2n)(2n+1)!) = (4n-1)!!/(2n+1) = A001147(2n)/(2n+1). - Valery A. Liskovets, Apr 13 2006

Extensions

More terms from Valery A. Liskovets, Apr 13 2006