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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.

%I A118446 #31 Oct 26 2024 12:07:29
%S A118446 21,1428,59136,1936935,55165110,1430857428,34701610944,800003272068,
%T A118446 17726513264460,380471504212800,7955313269904000,162738137109652650,
%U A118446 3267801532548762300,64578810084245919000,1258643138633207712000,24234564983959535297400,461636913607179055445700
%N 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.
%C A118446 Tree-rooted planar maps are counted by A005568 and tree-rooted maps on the torus by A118445.
%H A118446 E. A. Bender, E. R. Canfield and R. W. Robinson, <a href="https://doi.org/10.1016/0097-3165(88)90002-7">The asymptotic number of tree-rooted maps on a surface</a>, J. Comb. Theory, Ser. A, 48, No. 2 (1988), 156-164.
%H A118446 T. R. S. Walsh and A. B. Lehman, <a href="https://doi.org/10.1016/0095-8956(72)90049-4">Counting rooted maps by genus. II</a>, J. Comb. Theory, Ser. B, 13, No. 2 (1972), 122-141 (pp. 137, 140).
%F A118446 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.
%F A118446 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
%F A118446 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
%p A118446 C := proc(n) binomial(2*n,n)/(n+1) end:
%p A118446 b := proc(n) options remember;
%p A118446   if n<4 then 0 elif n=4 then 21 else
%p A118446     ((5*(n-1)+3)*(4*(n-1)+2)*b(n-1))/((5*(n-1)-2)*(n-1-3))
%p A118446   fi
%p A118446 end:
%p A118446 seq(add(binomial(2*n,2*i)*C(i)*b(n-i), i=0..n), n=4..20);
%p A118446 # _Mark van Hoeij_, Apr 06 2013
%t A118446 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)! );
%t A118446 Table[a[n], {n, 4, 20}] (* _Jean-François Alcover_, Aug 28 2019 *)
%o A118446 (PARI)
%o A118446 C(n) = binomial(2*n, n)/(n+1);
%o A118446 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))));
%o A118446 b(n)=A006298(n);
%o A118446 a(n)=sum(k=0,n, binomial(2*n,2*k) * C(k) * b(n-k) );
%o A118446 /* _Joerg Arndt_, Apr 07 2013 */
%K A118446 nonn
%O A118446 4,1
%A A118446 _Valery A. Liskovets_, May 04 2006
%E A118446 Corrected (replaced 34385678184 by 34701610944) and added more terms, _Mark van Hoeij_ and _Joerg Arndt_, Apr 07 2013