OFFSET
0,2
COMMENTS
Equivalently, the number of rooted quadrangulations of oriented surfaces with n quadrangles (and thus 2*n edges) for n > 0.
Equivalently, the number of pairs (alpha,sigma) of permutations on a pointed set of size 4*n up to simultaneous conjugacy such that alpha (resp. sigma) has only cycles of length 2 (resp. 4) and the subgroup generated by them acts transitively, for n > 0.
This is also the S(4, -6, 1) sequence of Martin and Kearney, if the offset is set to 1.
This sequence is not D-finite (or holonomic).
LINKS
Sasha Kolpakov, Table of n, a(n) for n = 0..119
Laura Ciobanu and Alexander Kolpakov, Free subgroups of free products and combinatorial hypermaps, Discrete Mathematics, 342 (2019), 1415-1433; arXiv:1708.03842 [math.CO], 2017-2019.
Bartomeu Fiol, Elena Gijon, and Unai Lejarza Alonso, Schwinger-Dyson approximants, arXiv:2511.05665 [hep-th], 2025. See p. 26, references.
R. J. Martin and M. J. Kearney, An exactly solvable self-convolutive recurrence, Aequat. Math., 80 (2010), 291-318; arXiv:1103.4936 [math.CO], 2011.
A. N. Stokes, Continued fraction solutions of the Riccati equation, Bull. Austral. Math. Soc. Vol. 25 (1982), 207-214.
FORMULA
a(0)=1, a(1)=3, a(n) = 4*n*a(n-1) + Sum_{k=1..n-2} a(k)*a(n-k-1) for n>=2.
From Peter Bala, Aug 22 2023: (Start)
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*n-1)*x/(1 - ... ))))))). (End)
PROG
(Python)
from sympy.core.cache import cacheit
@cacheit
def a(n): return 1 if n == 0 else (4*n - 2)*a(n - 1) + sum([a(k)*a(n - k - 1) for k in range(n)])
[a(n) for n in range(21)]
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Sasha Kolpakov, Sep 11 2017
EXTENSIONS
Edited by Andrey Zabolotskiy, Jan 23 2025
STATUS
approved