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A005701 Number of exterior points formed by extending diagonals of n-gon in general position.

%I A005701 M2968 #38 May 17 2025 05:53:40
%S A005701 3,14,40,90,175,308,504,780,1155,1650,2288,3094,4095,5320,6800,8568,
%T A005701 10659,13110,15960,19250,23023,27324,32200,37700,43875,50778,58464,
%U A005701 66990,76415,86800,98208,110704,124355
%N A005701 Number of exterior points formed by extending diagonals of n-gon in general position.
%C A005701 See Gouyou-Beauchamps for an interpretation in terms of closed paths in the first quadrant of the square grid.
%D A005701 Louis Comtet, Advanced Combinatorics, Reidel, 1974, p. 74, Problem 8.
%D A005701 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H A005701 Vincenzo Librandi, <a href="/A005701/b005701.txt">Table of n, a(n) for n = 0..1000</a>
%H A005701 Dominique Gouyou-Beauchamps, <a href="http://dx.doi.org/10.1007/BFb0072513">Chemins sous-diagonaux et tableaux de Young</a>, pp. 112-125 of "Combinatoire Enumerative (Montreal 1985)", Lect. Notes Math. 1234, Springer, 1986.
%H A005701 Dominique Gouyou-Beauchamps, <a href="/A005700/a005700.pdf">Chemins sous-diagonaux et tableaux de Young</a>, pp. 112-125 of "Combinatoire Enumerative (Montreal 1985)", Lect. Notes Math. 1234, Springer, 1986. (Annotated scanned copy)
%H A005701 <a href="/index/Y#Young">Index entries for sequences related to Young tableaux</a>.
%F A005701 a(n) = (n+1)*(n+2)*(n+3)*(n+6)/12.
%F A005701 G.f.: (x-3)/(x-1)^5. - Maksym Voznyy (voznyy(AT)mail.ru), Aug 10 2009
%F A005701 From _Amiram Eldar_, May 17 2025: (Start)
%F A005701 Sum_{n>=0} 1/a(n) = 137/300.
%F A005701 Sum_{n>=0} (-1)^n/a(n) = 32*log(2)/5 - 1247/300. (End)
%t A005701 CoefficientList[Series[(x - 3) / (x - 1)^5, {x, 0, 50}], x] (* _Vincenzo Librandi_, Jun 09 2013 *)
%o A005701 (Magma) [(n+1)*(n+2)*(n+3)*(n+6)/12: n in [0..50]]; // _Vincenzo Librandi_, Jun 09 2013
%o A005701 (PARI) a(n) = (n+1)*(n+2)*(n+3)*(n+6)/12; \\ _Michel Marcus_, Dec 16 2017
%Y A005701 A diagonal of the triangle in A179898.
%K A005701 nonn,easy
%O A005701 0,1
%A A005701 _N. J. A. Sloane_, _Robert G. Wilson v_