A118446 Number of tree-rooted maps of genus 2 with n edges: rooted maps with a distinguished spanning tree on an orientable surface of genus 2.
21, 1428, 59136, 1936935, 55165110, 1430857428, 34701610944, 800003272068, 17726513264460, 380471504212800, 7955313269904000, 162738137109652650, 3267801532548762300, 64578810084245919000, 1258643138633207712000, 24234564983959535297400, 461636913607179055445700
Offset: 4
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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Maple
C := proc(n) binomial(2*n,n)/(n+1) end: b := proc(n) options remember; if n<4 then 0 elif n=4 then 21 else ((5*(n-1)+3)*(4*(n-1)+2)*b(n-1))/((5*(n-1)-2)*(n-1-3)) fi end: seq(add(binomial(2*n,2*i)*C(i)*b(n-i), i=0..n), n=4..20); # Mark van Hoeij, Apr 06 2013 -
Mathematica
a[n_] := 2^(4n-9)(n-2)(5n^2+n+6) Gamma[n-3/2] Gamma[n+1/2]/(45 Pi (n-4)! (n+1)! ); Table[a[n], {n, 4, 20}] (* Jean-François Alcover, Aug 28 2019 *) -
PARI
C(n) = binomial(2*n, n)/(n+1); A006298(n) = if(n<4,0,if(n==4,21,((5*(n-1)+3)*(4*(n-1)+2)*A006298(n-1))/((5*(n-1)-2)*((n-1)-3)))); b(n)=A006298(n); a(n)=sum(k=0,n, binomial(2*n,2*k) * C(k) * b(n-k) ); /* Joerg Arndt, Apr 07 2013 */
Formula
a(n) = sum(k=0..n, binomial(2*n,2*k) * C(k) * b(n-k) ), where C(n)=A000108(n) - n-th Catalan number and b(n)=A006298(n) - the number of one-vertex maps of genus 2 for n>=4 and b(n)=0 for n<4.
G.f.: 7*x^4*(3*(1-9*x)*hypergeom([7/2,11/2],[6],16*x)+77*(1-6*x)*x*hypergeom([9/2,13/2],[7],16*x)). - Mark van Hoeij, Apr 07 2013
a(n) = (n-3)*(n-2)^2*(n-1)*n*(5*n^2+n+6) * binomial(2*n,n)^2 / (5760*(n+1)*(2*n-3)*(2*n-1)). - Vaclav Kotesovec, Oct 26 2024
Extensions
Corrected (replaced 34385678184 by 34701610944) and added more terms, Mark van Hoeij and Joerg Arndt, Apr 07 2013
Comments