A118445 - OEIS

A118445

Number of tree-rooted maps of genus 1 with n edges: rooted maps on the torus with a distinguished spanning tree.

1, 25, 490, 8820, 152460, 2576574, 42942900, 709171320, 11636856660, 190068658780, 3093732938296, 50222937310000, 813611584422000, 13158602740363500, 212528020730913000, 3428785401125396400, 55266606794455402500, 890117467077758188500

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OFFSET

2,2

COMMENTS

Tree-rooted planar maps are counted by A005568 and tree-rooted maps of (orientable) genus 2 by A118446. Typically, a(11) = 190068658780 = 2^2*5*7^2*11*13^2*17^2*19^2.

LINKS

Table of n, a(n) for n=2..19.

E. A. Bender, E. R. Canfield and R. W. Robinson, The asymptotic number of tree-rooted maps on a surface, J. Comb. Theory, Ser. A, 48, No. 2 (1988), 156-164.

T. R. S. Walsh and A. B. Lehman, Counting rooted maps by genus. II, J. Comb. Theory, Ser. B, 13, No. 2 (1972), 122-141 (pp. 137, 140).

FORMULA

a(n) = binomial(2n, 0) C(0) b(n) + binomial(2n, 2) C(1) b(n-1) + binomial(2n, 4) C(2) b(n-2) + ... + binomial(2n, 2n) C(n) b(0), where C(n) = A000108(n) - n-th Catalan number and b(n) = (2n-1)!/(6(n-2)! (n-1)!) = A002802(n-2) - the number of toroidal one-vertex maps with n edges for n >= 2 and b(0) = b(1) = 0.

O.g.f.: x^2 * hypergeom([5/2, 5/2], [4], 16*x). - Mark van Hoeij, Apr 06 2013

D-finite with recurrence -(n+1)*(n-2)*a(n) +4*((2*n-1)^2)*a(n-1)=0. - R. J. Mathar, Jul 27 2022

From Vaclav Kotesovec, Feb 17 2024: (Start)

a(n) = n*(n-1) * binomial(2*n,n)^2 / (24*(n+1)).

a(n) ~ 2^(4*n-3)/(3*Pi). (End)

MATHEMATICA

HypergeometricPFQ[{5/2, 5/2}, {4}, 16x] + O[x]^18 // CoefficientList[#, x]& (* Jean-François Alcover, Aug 28 2019 *)

Table[n*(n-1) * Binomial[2*n, n]^2 / (24*(n+1)), {n, 2, 20}] (* Vaclav Kotesovec, Feb 17 2024 *)

CROSSREFS

Sequence in context: A014927 A059946 A357147 * A000497 A353116 A028341

Adjacent sequences: A118442 A118443 A118444 * A118446 A118447 A118448

KEYWORD

nonn

AUTHOR

Valery A. Liskovets, May 04 2006

EXTENSIONS

Added more terms, Joerg Arndt, Apr 07 2013

STATUS

approved