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A006471 Number of tree-rooted planar maps with 4 faces and n vertices and no isthmuses.

%I A006471 M4021 #29 Aug 20 2025 12:15:32
%S A006471 5,84,650,3324,13020,42240,118998,300300,693693,1490060,3011580,
%T A006471 5779592,10608000,18728832,31957620,52907400,85261341,134115300,
%U A006471 206402966,311417700,461446700,672534720,965396250,1366496820,1909325925,2635885980,3598423704,4861432400,6503955744,8622225920
%N A006471 Number of tree-rooted planar maps with 4 faces and n vertices and no isthmuses.
%D A006471 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H A006471 Andrew Howroyd, <a href="/A006471/b006471.txt">Table of n, a(n) for n = 1..1000</a>
%H A006471 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 A006471 <a href="/index/Rec#order_10">Index entries for linear recurrences with constant coefficients</a>, signature (10,-45,120,-210,252,-210,120,-45,10,-1).
%F A006471 From _Colin Barker_, Apr 09 2013: (Start)
%F A006471 a(n) = (n*(2+n)^2*(3240 + 10158*n + 11777*n^2 + 6400*n^3 + 1770*n^4 + 242*n^5 + 13*n^6))/60480.
%F A006471 G.f.: x*(4*x^3 + 35*x^2 + 34*x + 5) / (x-1)^10. (End)
%t A006471 A006471[n_] := n*(n + 1)*(n + 2)^2*(n + 3)*(n + 4)*(n + 5)*(n*(13*n + 73) + 54)/60480;
%t A006471 Array[A006471, 50] (* _Paolo Xausa_, Aug 20 2025 *)
%o A006471 (PARI) a(n) = (n*(2+n)^2*(3240 +10158*n +11777*n^2 +6400*n^3 +1770*n^4 +242*n^5 +13*n^6))/60480 \\ _Andrew Howroyd_, Apr 03 2021
%Y A006471 Column 4 of A342987.
%K A006471 nonn
%O A006471 1,1
%A A006471 _N. J. A. Sloane_
%E A006471 Name clarified and terms a(12) and beyond from _Andrew Howroyd_, Apr 03 2021