A006298 -id:A006298 - cp's OEIS frontend

A288090 a(n) is the number of rooted maps with n edges and 10 faces on an orientable surface of genus 2.

7808250450, 955708437684, 56532447160536, 2200626948631386, 64232028100704156, 1511718920778951024, 30044423965980553536, 520516978029736518606, 8044640800289827566756, 112860842135424498808968, 1456882832375987896763184, 17491588653334242501297012, 197038603477850885815215480

Offset: 13

Gheorghe Coserea, Jun 05 2017

nonn

Sean R. Carrell, Guillaume Chapuy, Simple recurrence formulas to count maps on orientable surfaces, arXiv:1402.6300 [math.CO], 2014.

Rooted maps of genus 2 with n edges and f faces for 1<=f<=10: A006298 f=1, A288082 f=2, A288083 f=3, A288084 f=4, A288085 f=5, A288086 f=6, A288087 f=7, A288088 f=8, A288089 f=9, this sequence.
Column 10 of A269922.
Cf. A000108.

Q[0, 1, 0] = 1; Q[n_, f_, g_] /; n < 0 || f < 0 || g < 0 = 0;
Q[n_, f_, g_] := Q[n, f, g] = 6/(n + 1) ((2 n - 1)/3 Q[n - 1, f, g] + (2 n - 1)/3 Q[n - 1, f - 1, g] + (2 n - 3) (2 n - 2) (2 n - 1)/12 Q[n - 2, f, g - 1] + 1/2 Sum[l = n - k; Sum[v = f - u; Sum[j = g - i; Boole[l >= 1 && v >= 1 && j >= 0] (2 k - 1) (2 l - 1) Q[k - 1, u, i] Q[l - 1, v, j], {i, 0, g}], {u, 1, f}], {k, 1, n}]);
a[n_] := Q[n, 10, 2];
Table[a[n], {n, 13, 25}] (* Jean-François Alcover, Oct 18 2018 *)

A000108_ser(N) = my(x='x+O('x^(N+1))); (1 - sqrt(1-4*x))/(2*x);
A288090_ser(N) = {
  my(y = A000108_ser(N+1));
  6*y*(y-1)^13*(197300616213*y^12 + 2233379349250*y^11 + 1077980722075*y^10 - 16537713992125*y^9 + 7856375825902*y^8 + 29387232350368*y^7 - 33290642716432*y^6 + 994024496848*y^5 + 14078465181600*y^4 - 6737013421440*y^3 + 532103069696*y^2 + 244607984896*y - 34798091776)/(y-2)^38;
};
Vec(A288090_ser(13))

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.

Original entry on oeis.org

21, 1428, 59136, 1936935, 55165110, 1430857428, 34701610944, 800003272068, 17726513264460, 380471504212800, 7955313269904000, 162738137109652650, 3267801532548762300, 64578810084245919000, 1258643138633207712000, 24234564983959535297400, 461636913607179055445700

Offset: 4

Valery A. Liskovets, May 04 2006

nonn

Tree-rooted planar maps are counted by A005568 and tree-rooted maps on the torus by A118445.

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).

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

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 *)

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 */

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

Corrected (replaced 34385678184 by 34701610944) and added more terms, Mark van Hoeij and Joerg Arndt, Apr 07 2013

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A288090 a(n) is the number of rooted maps with n edges and 10 faces on an orientable surface of genus 2.

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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.

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