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Number of loopless tree-rooted planar maps with 4 vertices and n faces.
(Formerly M3716)
2

%I M3716 #21 Aug 01 2021 14:53:03

%S 0,4,135,1368,7350,28400,89073,241220,585057,1301420,2699125,5282172,

%T 9842430,17584416,30289835,50530680,81940901,129557940,200246795,

%U 303220720,450674190,658545360,947426925,1343646044,1880535825,2599922780,3553856649,4806611060

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

%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

%H Andrew Howroyd, <a href="/A006429/b006429.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_10">Index entries for linear recurrences with constant coefficients</a>, signature (10,-45,120,-210,252,-210,120,-45,10,-1).

%F 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 From _Chai Wah Wu_, Aug 01 2021: (Start)

%F 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 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)

%o (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 Column 4 of A342985.

%K nonn

%O 1,2

%A _N. J. A. Sloane_

%E Title improved by _Sean A. Irvine_, Apr 10 2017

%E Terms a(12) and beyond from _Andrew Howroyd_, Apr 03 2021