cp's OEIS Frontend

A292186 Number of rooted unlabeled connected four-regular maps on a compact closed oriented surface with n vertices (and thus 2*n edges), with a(0) = 1.

%I A292186 #53 Jan 30 2025 05:08:34
%S A292186 1,3,24,297,4896,100278,2450304,69533397,2247492096,81528066378,
%T A292186 3280382613504,145009234904922,6986546222800896,364418301804218028,
%U A292186 20459842995693256704,1230262900677124568397,78884016707711348637696,5372823210133041283250178,387394283866652086938107904
%N A292186 Number of rooted unlabeled connected four-regular maps on a compact closed oriented surface with n vertices (and thus 2*n edges), with a(0) = 1.
%C A292186 Equivalently, the number of rooted quadrangulations of oriented surfaces with n quadrangles (and thus 2*n edges) for n > 0.
%C A292186 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.
%C A292186 This is also the S(4, -6, 1) sequence of Martin and Kearney, if the offset is set to 1.
%C A292186 This sequence is not D-finite (or holonomic).
%H A292186 Sasha Kolpakov, <a href="/A292186/b292186.txt">Table of n, a(n) for n = 0..119</a>
%H A292186 Laura Ciobanu and Alexander Kolpakov, <a href="https://doi.org/10.1016/j.disc.2019.01.014">Free subgroups of free products and combinatorial hypermaps</a>, Discrete Mathematics, 342 (2019), 1415-1433; arXiv:<a href="https://arxiv.org/abs/1708.03842">1708.03842</a> [math.CO], 2017-2019.
%H A292186 R. J. Martin and M. J. Kearney, <a href="https://doi.org/10.1007/s00010-010-0051-0">An exactly solvable self-convolutive recurrence</a>, Aequat. Math., 80 (2010), 291-318; arXiv:<a href="https://arxiv.org/abs/1103.4936">1103.4936</a> [math.CO], 2011.
%H A292186 A. N. Stokes, <a href="https://doi.org/10.1017/S0004972700005219">Continued fraction solutions of the Riccati equation</a>, Bull. Austral. Math. Soc. Vol. 25 (1982), 207-214.
%F A292186 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.
%F A292186 From _Peter Bala_, Aug 22 2023: (Start)
%F A292186 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.
%F A292186 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 - ... )))))).
%F A292186 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)
%o A292186 (Python)
%o A292186 from sympy.core.cache import cacheit
%o A292186 @cacheit
%o A292186 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)])
%o A292186 [a(n) for n in range(21)]
%Y A292186 Column 4 of A380622.
%Y A292186 Cf. A172455, A292187, A128709.
%K A292186 nonn,easy
%O A292186 0,2
%A A292186 _Sasha Kolpakov_, Sep 11 2017
%E A292186 Edited by _Andrey Zabolotskiy_, Jan 23 2025