OFFSET
4,1
COMMENTS
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).
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
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 *)
PROG
(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 */
CROSSREFS
KEYWORD
nonn
AUTHOR
Valery A. Liskovets, May 04 2006
EXTENSIONS
Corrected (replaced 34385678184 by 34701610944) and added more terms, Mark van Hoeij and Joerg Arndt, Apr 07 2013
STATUS
approved