A090372
Number of unrooted planar 3-constellations with n triangles.
Original entry on oeis.org
1, 6, 22, 174, 1479, 16808, 201834, 2631594, 35965555, 512062566, 7528425420, 113708935808, 1756853846316, 27676951028496, 443411345677658, 7209139541742750, 118738765611199983, 1978360119497335826
Offset: 1
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with(numtheory): C_3 := proc(n) local s,d; if n=0 then RETURN(1) else s := -3^n*binomial(3*n,n); for d in divisors(n) do s := s+phi(n/d)*3^d*binomial(3*d,d) od; RETURN((4/(3*n))*(3^n*binomial(3*n,n)/((2*n+1)*(2*n+2))+s/2)); fi; end;
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a[0] = 1; a[n_] := Module[{s, d}, s = -3^n Binomial[3n, n]; Do[s = s + EulerPhi[n/d] 3^d Binomial[3d, d], {d, Divisors[n]}]; (4/(3n)) (3^n Binomial[3n, n]/((2n+1)(2n+2)) + s/2)];
Array[a, 18] (* Jean-François Alcover, Jul 24 2018, from Maple *)
A090374
Number of rooted planar 4-constellations with n quadrangles: rooted planar maps with bicolored faces having n black quadrangular faces and an arbitrary number of white faces of degrees multiple to 4.
Original entry on oeis.org
1, 10, 160, 3200, 72960, 1813504, 47923200, 1325629440, 37991219200, 1120005652480, 33789432561664, 1039157228994560, 32480974549811200, 1029463445864448000, 33023079530417356800, 1070513886720329515008, 35026358912891580579840, 1155516042520241436098560
Offset: 1
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A090374 := proc(n)
5*4^(n-1)*binomial(4*n, n)/((3*n+1)*(3*n+2))
end proc:
seq(A090374(n),n=1..40) ; # R. J. Mathar, Mar 29 2023
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a[n_] := 5 2^(2n) (4n-1)! / ((n-1)! (3n+2)!);
Array[a, 18] (* Jean-François Alcover, Aug 28 2019 *)
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vector(20, n, 5*4^(n-1)*binomial(4*n, n)/((3*n+1)*(3*n+2))) \\ Michel Marcus, Dec 11 2014
A298358
a(n) is the number of rooted 3-connected bicubic planar maps with 2n vertices.
Original entry on oeis.org
1, 0, 0, 1, 0, 3, 7, 15, 63, 168, 561, 1881, 6110, 21087, 72174, 250775, 883116, 3125910, 11174280, 40209852, 145590720, 530358095, 1941862860, 7144623447, 26403493545, 97971775008, 364903633215, 1363847131450, 5113975285788, 19233646581282
Offset: 1
A(x) = x + x^4 + 3*x^6 + 7*x^7 + 15*x^8 + 63*x^9 + 168*x^10 + 561*x^11 + ...
- Gheorghe Coserea, Table of n, a(n) for n = 1..501
- Daniel Birmajer, Juan B. Gil, Michael D. Weiner, A family of Bell transformations, arXiv:1803.07727 [math.CO], 2018.
- Hsien-Kuei Hwang, Mihyun Kang, Guan-Huei Duh, Asymptotic Expansions for Sub-Critical Lagrangean Forms, LIPIcs Proceedings of Analysis of Algorithms 2018, Vol. 110. Schloss Dagstuhl-Leibniz-Zentrum für Informatik, 2018.
- W. T. Tutte, A census of planar maps, Canad. J. Math., 15(1963), 249-271.
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kmax = 30; b[0] = 1; b[n_] := 3*2^(n - 1)*CatalanNumber[n]/(n + 2);
G[x_] = Sum[b[k] x^k, {k, 0, kmax}];
A[_] = 1;
Do[A[x_] = G[x/(1 + A[x] + O[x]^k)^3] - 1 // Normal, {k, 1, kmax + 1}];
CoefficientList[A[x], x][[2 ;; -2]] (* Jean-François Alcover, Jun 19 2018 *)
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seq(N) = {
my(x='x+O('x^(N+2)), y=(1-sqrt(1-8*x))/(4*x), g=(1+4*y-y^2)/4);
Vec(subst(g-1, 'x, serreverse(x*g^3)));
};
seq(30) \\ Gheorghe Coserea, Apr 11 2018
Showing 1-3 of 3 results.
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