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%I M2974 N1203 #31 Jan 01 2019 06:34:05
%S 1,1,3,14,80,518,3647,27274,213480,1731652,14455408,123552488,
%T 1077096124,9548805240,85884971043,782242251522,7203683481720,
%U 66989439309452,628399635777936,5940930064989720,56562734108608536
%N Number of 3-edge-connected rooted cubic maps with 2n nodes and a distinguished Hamiltonian cycle.
%D N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H Vincenzo Librandi, <a href="/A000264/b000264.txt">Table of n, a(n) for n = 1..200</a>
%H L. B. Richmond, <a href="http://dx.doi.org/10.1016/0095-8956(76)90031-9">On Hamiltonian polygons</a>, J. Combinatorial Theory Ser. B 21 (1976), no. 1, 81--87. MR0432491 (55 #5479) [See v_n].
%H W. T. Tutte, <a href="http://dx.doi.org/10.4153/CJM-1962-032-x">A census of Hamiltonian polygons</a>, Canad. J. Math., 14 (1962), 402-417.
%F Let b(n)=(2n)!*(2n+2)!/(2*n!*(n+1)!^2*(n+2)!). Let B(x) be the generating function producing b(n), and A(x) be the generating function producing a(n). Then these sequences satisfy the functional equation B(x)=A(x(1+2*B(x))^2). - _Sean A. Irvine_, Apr 05 2010
%t max = 21; b[n_] := (2n)!*(2n + 2)!/(2*n!*(n + 1)!^2*(n + 2)!); b[0] = 0; bf[x_] := Sum[b[n]*x^n, {n, 0, max}]; Clear[a]; a[0] = 0; a[1] = a[2] = 1; af[x_] := Sum[a[n]*x^n, {n, 0, max}]; se = Series[bf[x] - af[x*(1 + 2*bf[x])^2], {x, 0, max}] // Normal; Table[a[n], {n, 1, max}] /. SolveAlways[se == 0, x] // First (* _Jean-François Alcover_, Jan 31 2013, after _Sean A. Irvine_ *)
%Y Cf. A000309, A000356, A004304.
%K nonn,nice
%O 1,3
%A _N. J. A. Sloane_
%E Better definition from _Michael Albert_, Oct 24 2008
%E More terms from _Sean A. Irvine_, Apr 05 2010
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