OFFSET
1,2
LINKS
Evgeniy Krasko, Alexander Omelchenko, Enumeration of 4-regular one-face maps, European Journal of Combinatorics, Volume 62, 2017, Pages 167-177. Summation limit "floor(g/2+r_4/4)" in Formula (14) should read "floor(g/2-r_4/4)".
EXAMPLE
On a torus (g=1) there exists only one unrooted 4-regular one-face map. On a double torus (g=2) there are 6 such maps.
PROG
(Python 2.7)
import math
fact = math.factorial
def a(g):
s = 0
s += 2 * fact(4 * g - 2) / (4 ** g * fact(g) * fact(g - 1))
s += 3 * fact(4 * g - 2) / (fact (2 * g + 1) * fact(2 * g - 2))
s += sum((4 * g - 2) * fact(2 * g - 2 * g_ + k - 1) / (fact(2 * k - 4 * g_ + 2) * fact(g_) * fact(k - g_) * fact (2 * g - 1 - 2 * k)) for g_ in range(1, g / 2 + 1) for k in range(2 * g_ - 1, g))
s += sum((4 * g - 2) * 2 ** (2 * g_ - 2 + r4) * fact(k - 2 * g_ + g - r4 / 2) / (fact(g_) * fact(k - g_) * fact(g - r4 / 2 - 2 * k) * fact(2 * k + 3 - 4 * g_ - r4) * fact(r4 - 1)) for g_ in range(g / 4 + 1) for r4 in range(2, 2 * (g + 3 - 4 * g_) / 3 + 1, 2) for k in range(2 * g_ - 1 + r4 / 2, (2 * g - r4) / 4 + 1))
return s / (8 * g - 4)
CROSSREFS
KEYWORD
nonn
AUTHOR
Evgeniy Krasko, Sep 03 2017
STATUS
approved