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A006413 Number of nonseparable tree-rooted planar maps with n + 4 edges and 5 vertices.

%I A006413 M4031 #31 Aug 20 2025 09:23:41
%S A006413 5,210,3150,27556,170793,829920,3359356,11786190,36845718,104719524,
%T A006413 274707420,672982128,1554007910,3407724936,7139933088,14366348780,
%U A006413 27878652291,52364814150,95497666810,169546939380,293722986375,497527759560,825473130300,1343631834090
%N A006413 Number of nonseparable tree-rooted planar maps with n + 4 edges and 5 vertices.
%D A006413 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H A006413 Andrew Howroyd, <a href="/A006413/b006413.txt">Table of n, a(n) for n = 1..1000</a>
%H A006413 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 A006413 <a href="/index/Rec#order_14">Index entries for linear recurrences with constant coefficients</a>, signature (14,-91,364,-1001,2002,-3003,3432,-3003,2002,-1001,364,-91,14,-1).
%F A006413 a(n) = 5 * binomial(n + 6, 7) + 170 * binomial(n + 6, 8) + 1440 * binomial(n + 6, 9) + 4906 * binomial(n + 6, 10) + 7927 * binomial(n + 6, 11) + 6090 * binomial(n + 6, 12) + 1794 * binomial(n + 6, 13). - _Sean A. Irvine_, Apr 03 2017
%F A006413 a(n) = binomial(n+7,8)*(n + 4)*(23*n^4 + 279*n^3 + 941*n^2 + 599*n + 138)/1980. - _Andrew Howroyd_, Apr 05 2021
%F A006413 G.f.: x*(5 + 140*x + 665*x^2 + 746*x^3 + 224*x^4 + 14*x^5)/(1 - x)^14. - _Stefano Spezia_, Aug 19 2025
%t A006413 A006413[n_] := Binomial[n + 7, 8]*(n + 4)*(n*(n*(n*(23*n + 279) + 941) + 599) + 138)/1980;
%t A006413 Array[A006413, 25] (* _Paolo Xausa_, Aug 20 2025 *)
%o A006413 (PARI) a(n) = {binomial(n+7, 8)*(n + 4)*(23*n^4 + 279*n^3 + 941*n^2 + 599*n + 138)/1980} \\ _Andrew Howroyd_, Apr 05 2021
%Y A006413 Column 5 of A342984.
%Y A006413 Cf. A006411, A006412.
%K A006413 nonn,easy
%O A006413 1,1
%A A006413 _N. J. A. Sloane_
%E A006413 Terms a(10) and beyond from _Andrew Howroyd_, Apr 05 2021