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

0,3

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).

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

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

Prog

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

Crossrefs

Right-hand diagonal of A035309.

Cf. A035309.

Sequence in context: A298851 A118446 A130332 * A278323 A301432 A220561

Adjacent sequences: A035316 A035317 A035318 * A035320 A035321 A035322

Keyword

nonn

Author

N. J. A. Sloane

Extensions

More terms from Valery A. Liskovets, Apr 13 2006

Status

approved