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%I #41 Apr 15 2018 02:07:43
%S 1,1,2,8,54,442,4032,39706,413358,4487693,50348500,579994802,
%T 6827955072,81854670861,996529292432,12293898494952,153421680489694,
%U 1934041122204318,24599034335501730,315369011873625930,4072021557616191708,52915860528084306704,691646518495876375968
%N Number of rooted non-separable bi-Eulerian planar maps with 2n edges. Bi-Eulerian means all its vertices and faces are of even valency.
%C The formula from the article by Liskovets and Walsh, p. 218, B'ns(n), gives incorrect data {1, 4, 25, 204, 1964, 21070, 243681, ...}. Here is the incorrect formula rewritten into Mathematica: Table[(Sum[(-1)^j*3^(n - j - 1)*Binomial[2*n + j - 1, j] * Sum[(-1)^k*Binomial[j, k]*Binomial[3*n, n - j - k - 1], {k, 0, Min[j, n - j - 1]}], {j, 0, n - 1}] - 2*Sum[(-1)^j*3^(n - j - 1)*Binomial[2*n + j - 1, j] * Sum[(-1)^k*Binomial[j, k]*Binomial[3*n, n - j - k - 2], {k, 0, Min[j, n - j - 2]}], {j, 0, n - 2}])/n, {n, 1, 20}]. - _Vaclav Kotesovec_, Apr 13 2018
%H Gheorghe Coserea, <a href="/A069729/b069729.txt">Table of n, a(n) for n = 0..500</a>
%H V. A. Liskovets and T. R. S. Walsh, <a href="http://dx.doi.org/10.1016/j.disc.2003.09.015">Enumeration of Eulerian and unicursal planar maps</a>, Discr. Math., 282 (2004), 209-221.
%F G.f. y=A(x) satisfies 0 = y^9 - y^8 + 18*x*y^6 - 66*x*y^5 + 47*x*y^4 + 81*x^2*y^3 - 81*x^2*y^2 + 27*x^2*y - 3*x^2. - _Gheorghe Coserea_, Apr 13 2018
%F a(n) ~ 2^(6*n - 1) * 3^(8*n - 1/2) / (3125 * sqrt(Pi) * 13^(4*n - 5/2) * n^(5/2)). - _Vaclav Kotesovec_, Apr 13 2018
%F A(x) = 1 + serreverse(-(1+x)^4*(18*x^2-30*x-1 + (1-12*x)^(3/2))/(6*(3*x+2)^3)); equivalently, it can be rewritten as A(x) = 1 + serreverse((y - 1)*(3*y^2 + y - 1)^4 / (243 * y^6 * (2*y-1)^3)), where y = A000108(3*x). - _Gheorghe Coserea_, Apr 14 2018
%e A(x) = 1 + x + 2*x^2 + 8*x^3 + 54*x^4 + 442*x^5 + 4032*x^6 + ...
%t CoefficientList[1 + InverseSeries[Series[-(1+x)^4*(18*x^2-30*x-1 + (1-12*x)^(3/2))/(6*(3*x+2)^3), {x, 0, 25}], x], x] (* _Vaclav Kotesovec_, Apr 14 2018, after _Gheorghe Coserea_ *)
%o (PARI)
%o seq(N) = {
%o my(x='x+O('x^(2*N-1)), y=1+serreverse(x/(3*(1+x)^3)), f=(1+3*y-y^2)/3,
%o g=subst(f, 'x, 'x^2), v=Vec(subst(g, 'x, serreverse(x*g^2))));
%o vector((#v+1)\2, n, v[2*n-1]);
%o };
%o seq(23) \\ _Gheorghe Coserea_, Apr 13 2018
%Y Cf. A069726, A005470.
%K nonn
%O 0,3
%A _Valery A. Liskovets_, Apr 07 2002
%E More terms from _Gheorghe Coserea_, Apr 13 2018
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