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A006429 Number of loopless tree-rooted planar maps with 4 vertices and n faces.

%I A006429 M3716 #28 Aug 20 2025 18:11:32
%S A006429 0,4,135,1368,7350,28400,89073,241220,585057,1301420,2699125,5282172,
%T A006429 9842430,17584416,30289835,50530680,81940901,129557940,200246795,
%U A006429 303220720,450674190,658545360,947426925,1343646044,1880535825,2599922780,3553856649,4806611060
%N A006429 Number of loopless tree-rooted planar maps with 4 vertices and n faces.
%D A006429 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H A006429 Andrew Howroyd, <a href="/A006429/b006429.txt">Table of n, a(n) for n = 1..1000</a>
%H A006429 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 A006429 <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 A006429 a(n) = n*(n+2)*(13*n^7+268*n^6+2254*n^5+4900*n^4-10703*n^3-62048*n^2+28596*n+137520) / 60480 for n > 1. - _Sean A. Irvine_, Apr 10 2017
%F A006429 From _Chai Wah Wu_, Aug 01 2021: (Start)
%F A006429 a(n) = 10*a(n-1) - 45*a(n-2) + 120*a(n-3) - 210*a(n-4) + 252*a(n-5) - 210*a(n-6) + 120*a(n-7) - 45*a(n-8) + 10*a(n-9) - a(n-10) for n > 11.
%F A006429 G.f.: x^2*(-5*x^9 + 50*x^8 - 224*x^7 + 590*x^6 - 995*x^5 + 1100*x^4 - 735*x^3 + 198*x^2 + 95*x + 4)/(x - 1)^10. (End)
%t A006429 A006429[n_] := If[n == 1, 0, (n*(n + 2)*(n*(n*(n*(n*(n*(n*(13*n + 268) + 2254) + 4900) - 10703) - 62048) + 28596) + 137520))/60480];
%t A006429 Array[A006429, 50] (* _Paolo Xausa_, Aug 20 2025 *)
%o A006429 (PARI) a(n) = if(n < 2, 0, n*(n+2)*(13*n^7+268*n^6+2254*n^5+4900*n^4-10703*n^3-62048*n^2+28596*n+137520) / 60480) \\ _Andrew Howroyd_, Apr 03 2021
%Y A006429 Column 4 of A342985.
%K A006429 nonn,easy
%O A006429 1,2
%A A006429 _N. J. A. Sloane_
%E A006429 Title improved by _Sean A. Irvine_, Apr 10 2017
%E A006429 Terms a(12) and beyond from _Andrew Howroyd_, Apr 03 2021