cp's OEIS Frontend

A308523 Number of essentially simple rooted toroidal triangulations with n vertices.

%I A308523 #47 Jul 29 2025 08:40:22
%S A308523 0,1,10,97,932,8916,85090,810846,7719048,73431340,698187400,
%T A308523 6635738209,63047912372,598885073788,5687581936284,54005562798252,
%U A308523 512728901004816,4867263839614716,46199494669833400,438481077306427924,4161316466910824272
%N A308523 Number of essentially simple rooted toroidal triangulations with n vertices.
%H A308523 Michael De Vlieger, <a href="/A308523/b308523.txt">Table of n, a(n) for n = 0..499</a>
%H A308523 Nicolas Bonichon, Éric Fusy, Benjamin Lévêque, <a href="https://arxiv.org/abs/1907.04016">A bijection for essentially 3-connected toroidal maps</a>, arXiv:1907.04016 [math.CO], 2019.
%H A308523 Éric Fusy, Benjamin Lévêque, <a href="https://arxiv.org/abs/1807.00522">Orientations and bijections for toroidal maps with prescribed face-degrees and essential girth</a>, arXiv:1807.00522 [math.CO], 2018. See Proposition 25 p. 37.
%F A308523 G.f.: A/(1-3*A)^2 where A=x(1+A)^4 is the g.f. of A002293.
%F A308523 From _Vaclav Kotesovec_, Jun 25 2019: (Start)
%F A308523 Recurrence: 81*(n-1)*(3*n - 2)*(3*n - 1)*(24*n - 37)*a(n) = 24*(13824*n^4 - 59328*n^3 + 92832*n^2 - 62278*n + 14653)*a(n-1) - 2048*(2*n - 3)*(4*n - 7)*(4*n - 5)*(24*n - 13)*a(n-2).
%F A308523 a(n) ~ 2^(8*n - 3) / 3^(3*n). (End)
%F A308523 From _Seiichi Manyama_, Jul 19 2025: (Start)
%F A308523 G.f.: g*(1-g)/(1-4*g)^2 where g*(1-g)^3 = x.
%F A308523 L.g.f.: Sum_{k>=1} a(k)*x^k/k = (1/4) * log( Sum_{k>=0} binomial(4*k,k)*x^k ). (End)
%F A308523 From _Seiichi Manyama_, Jul 28 2025: (Start)
%F A308523 a(n) = Sum_{k=0..n-1} binomial(4*k-2+l,k) * binomial(4*n-4*k-l,n-k-1) for every real number l.
%F A308523 a(n) = Sum_{k=0..n-1} 3^(n-k-1) * binomial(4*n-1,k).
%F A308523 a(n) = Sum_{k=0..n-1} 4^(n-k-1) * binomial(3*n+k-1,k). (End)
%p A308523 n:=20:
%p A308523 dev_A := series(RootOf(A-x*(1+A)^4, A), x = 0, n+1):
%p A308523 seq(coeff(series(subs(A=dev_A, A/(1-3*A)^2), x, n+1), x, k), k=0..n);
%t A308523 terms = 21;
%t A308523 A[_] = 0; Do[A[x_] = x (1 + A[x])^4 + O[x]^terms, terms];
%t A308523 CoefficientList[A[x]/(1 - 3 A[x])^2, x] (* _Jean-François Alcover_, Jun 17 2019 *)
%o A308523 (PARI) my(N=30, x='x+O('x^N), g=x*sum(k=0, N, binomial(4*k+2, k)/(k+1)*x^k)); concat(0, Vec(g*(1-g)/(1-4*g)^2)) \\ _Seiichi Manyama_, Jul 19 2025
%Y A308523 Cf. A002293, A308524, A308526, A289208, A006422.
%Y A308523 Cf. A000302, A036829.
%Y A308523 Cf. A386565, A386611, A386612.
%K A308523 nonn
%O A308523 0,3
%A A308523 _Nicolas Bonichon_, Jun 05 2019