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A006412 Number of nonseparable tree-rooted planar maps with n + 3 edges and 4 vertices.

%I A006412 M3697 #29 Aug 20 2025 07:40:27
%S A006412 4,75,604,3150,12480,40788,115500,292578,677820,1459315,2954952,
%T A006412 5679700,10438272,18449760,31511880,52213596,84206100,132543411,
%U A006412 204105220,308116050,456776320,666022500,956435220,1354315950,1892954700,2614113099,3569749200,4824012424
%N A006412 Number of nonseparable tree-rooted planar maps with n + 3 edges and 4 vertices.
%D A006412 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H A006412 Andrew Howroyd, <a href="/A006412/b006412.txt">Table of n, a(n) for n = 1..1000</a>
%H A006412 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 A006412 <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 A006412 a(n) = 4 * binomial(n + 4, 5) + 51 * binomial(n + 4, 6) + 163 * binomial(n + 4, 7) + 194 * binomial(n + 4, 8) + 78 * binomial(n + 4, 9). - _Sean A. Irvine_, Apr 03 2017
%F A006412 a(n) = binomial(n+5,6)*(n + 3)*(13*n^2 + 57*n + 14)/84. - _Andrew Howroyd_, Apr 05 2021
%F A006412 G.f.: x*(4 + 35*x + 34*x^2 + 5*x^3)/(1 - x)^10. - _Stefano Spezia_, Aug 19 2025
%t A006412 A006412[n_] := Binomial[n + 5, 6]*(n + 3)*(n*(13*n + 57) + 14)/84;
%t A006412 Array[A006412, 30] (* _Paolo Xausa_, Aug 20 2025 *)
%o A006412 (PARI) a(n) = {binomial(n+5,6)*(n + 3)*(13*n^2 + 57*n + 14)/84} \\ _Andrew Howroyd_, Apr 05 2021
%Y A006412 Column 4 of A342984.
%Y A006412 Cf. A006411, A006413.
%K A006412 nonn,easy
%O A006412 1,1
%A A006412 _N. J. A. Sloane_
%E A006412 Terms a(11) and beyond from _Andrew Howroyd_, Apr 05 2021