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%I M3139 #24 Aug 08 2022 21:55:58
%S 0,3,36,135,360,798,1568,2826,4770,7645,11748,17433,25116,35280,48480,
%T 65348,86598,113031,145540,185115,232848,289938,357696,437550,531050,
%U 639873,765828,910861,1077060,1266660,1482048,1725768,2000526,2309195,2654820,3040623
%N Number of loopless tree-rooted planar maps with 3 vertices and n faces and no isthmuses.
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H Andrew Howroyd, <a href="/A006428/b006428.txt">Table of n, a(n) for n = 1..1000</a>
%H T. R. S. Walsh and A. B. Lehman, <a href="http://dx.doi.org/10.1016/0095-8956(75)90050-7">Counting rooted maps by genus. III: Nonseparable maps</a>, J. Combinatorial Theory Ser. B 18 (1975), 222-259.
%H <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (6,-15,20,-15,6,-1).
%F a(n) seems to be divisible by n+1. - _Ralf Stephan_, Sep 01 2003
%F Conjecture (for n > 1): a(n) = n*(n+1)*(n^3+6*n^2+11*n-42) / 24. - _Sean A. Irvine_, Apr 10 2017
%F The above conjectures are true. - _Andrew Howroyd_, Apr 03 2021
%F From _Chai Wah Wu_, Aug 08 2022: (Start)
%F a(n) = 6*a(n-1) - 15*a(n-2) + 20*a(n-3) - 15*a(n-4) + 6*a(n-5) - a(n-6) for n > 7.
%F G.f.: x^2*(2*x^5 - 12*x^4 + 30*x^3 - 36*x^2 + 18*x + 3)/(x - 1)^6. (End)
%o (PARI) a(n) = if(n<2, 0, n*(n+1)*(n^3+6*n^2+11*n-42) / 24) \\ _Andrew Howroyd_, Apr 03 2021
%Y Column 3 of A342985.
%K nonn,easy
%O 1,2
%A _N. J. A. Sloane_
%E Title improved by _Sean A. Irvine_, Apr 10 2017
%E Terms a(13) and beyond from _Andrew Howroyd_, Apr 03 2021
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