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A006468 Number of rooted planar maps with 4 faces and n vertices and no isthmuses.

%I A006468 M3984 #37 Aug 21 2025 00:12:34
%S A006468 5,37,150,449,1113,2422,4788,8790,15213,25091,39754,60879,90545,
%T A006468 131292,186184,258876,353685,475665,630686,825517,1067913,1366706,
%U A006468 1731900,2174770,2707965,3345615,4103442,4998875,6051169,7281528,8713232,10371768,12284965,14483133,16999206
%N A006468 Number of rooted planar maps with 4 faces and n vertices and no isthmuses.
%D A006468 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H A006468 Andrew Howroyd, <a href="/A006468/b006468.txt">Table of n, a(n) for n = 1..1000</a>
%H A006468 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 A006468 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, Table VIb, f=5.
%H A006468 <a href="/index/Rec#order_07">Index entries for linear recurrences with constant coefficients</a>, signature (7,-21,35,-35,21,-7,1).
%F A006468 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 A006468 The above conjecture is true. - _Andrew Howroyd_, Apr 02 2021
%t A006468 A006468[n_] := n*(n + 1)*(n + 2)*(n*(n*(2*n + 33) + 142) + 123)/360;
%t A006468 Array[A006468, 50] (* _Paolo Xausa_, Aug 20 2025 *)
%o A006468 (PARI) a(n) = {n *(n+1) *(n+2) *(2*n^3 + 33*n^2 + 142*n + 123) /360} \\ _Andrew Howroyd_, Apr 02 2021
%Y A006468 Column k=4 of A342981.
%K A006468 nonn
%O A006468 1,1
%A A006468 _N. J. A. Sloane_
%E A006468 Title improved and a(13)-a(14) from _Sean A. Irvine_, Apr 24 2017
%E A006468 Terms a(15) and beyond from _Andrew Howroyd_, Apr 02 2021