

A292186


Number of rooted unlabeled connected fourregular maps on a compact closed oriented surface with n vertices (and thus 2*n edges).


4



1, 3, 24, 297, 4896, 100278, 2450304, 69533397, 2247492096, 81528066378, 3280382613504, 145009234904922, 6986546222800896, 364418301804218028, 20459842995693256704, 1230262900677124568397, 78884016707711348637696, 5372823210133041283250178, 387394283866652086938107904
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OFFSET

0,2


COMMENTS

Equivalently, the number of rooted quadrangulations of oriented surfaces with n quadrangles (and thus 2*n edges).
Equivalently, the number of pairs of permutations (alpha,sigma) up to simultaneous conjugacy on a pointed set of size 4*n with alpha^2=sigma^4=1, acting transitively and without fixed points.
This is also the S(4, 6, 1) sequence of Martin and Kearney.
This sequence is not Dfinite (or holonomic).


LINKS



FORMULA

a(0)=1, a(1)=3, a(n) = 4*n*a(n1) + Sum_{k=1..n2} a(k)*a(nk1) for n>=2.
The o.g.f. A(x) = 1 + 3*x + 24*x^2 + 297*x^3 + 4896*x^4 + 100278*x^5 + 2450304*x^6 + ... satisfies the Riccati differential equation (4*x^2)*A'(x) = 1 + (1  2*x)*A(x)  x*A(x)^2 with A(0) = 1.
O.g.f. as a continued fraction of Stieltjes type: 1/(1  3*x/(1  5*x/(1  7*x/(1  9*x/(1  ...  (2*n+1)*x/(1  ... )))))).
Also A(x) = 1/(1 + 2*x  5*x/(1  3*x/(1  9*x/(1  7*x/(1  ...  (4*n+1)*x/(1  (4*n1)*x/(1  ... ))))))). (End)


PROG

(Python)
from sympy.core.cache import cacheit
@cacheit
def a(n): return n*2 + 1 if n < 2 else 4*n*a(n  1) + sum([a(k)*a(n  k  1) for k in range(1, n  1)])
[a(n) for n in range(21)]


CROSSREFS



KEYWORD

nonn,easy


AUTHOR



STATUS

approved



