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Number of rooted planar maps with 4 faces and n vertices and no isthmuses.
(Formerly M3984)
3

%I M3984 #30 Apr 02 2021 19:55:26

%S 5,37,150,449,1113,2422,4788,8790,15213,25091,39754,60879,90545,

%T 131292,186184,258876,353685,475665,630686,825517,1067913,1366706,

%U 1731900,2174770,2707965,3345615,4103442,4998875,6051169,7281528,8713232,10371768,12284965,14483133,16999206

%N Number of rooted planar maps with 4 faces and n vertices 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="/A006468/b006468.txt">Table of n, a(n) for n = 1..1000</a>

%H Simon Plouffe, <a href="http://arxiv.org/abs/0911.4975">Approximations of generating functions and a few conjectures</a>, Master's thesis, UQAM, 1992; arXiv:0911.4975 [math.NT], 2009.

%H Walsh, T. R. S.; Lehman, A. B.; <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, Table VIb, f=5.

%F G.f.: -x*(x^3-4*x^2+2*x+5)/(x-1)^7, equivalent to a(n) = n *(n+1) *(n+2) *(2*n^3 +33*n^2 +142*n +123) /360, conjectured in _Simon Plouffe_'s Master's thesis, 1992.

%F The above conjecture is true. - _Andrew Howroyd_, Apr 02 2021

%o (PARI) a(n) = {n *(n+1) *(n+2) *(2*n^3 + 33*n^2 + 142*n + 123) /360} \\ _Andrew Howroyd_, Apr 02 2021

%Y Column k=4 of A342981.

%K nonn

%O 1,1

%A _N. J. A. Sloane_

%E Title improved and a(13)-a(14) from _Sean A. Irvine_, Apr 24 2017

%E Terms a(15) and beyond from _Andrew Howroyd_, Apr 02 2021