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
Offset: 2
Keywords
Links
- 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).
Programs
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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 *)
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)
Extensions
Added more terms, Joerg Arndt, Apr 07 2013
Comments