A117662 a(n) = n*(n-1)*(n-2)*(n+3)/12.

0, 0, 0, 3, 14, 40, 90, 175, 308, 504, 780, 1155, 1650, 2288, 3094, 4095, 5320, 6800, 8568, 10659, 13110, 15960, 19250, 23023, 27324, 32200, 37700, 43875, 50778, 58464, 66990, 76415, 86800, 98208, 110704, 124355, 139230, 155400, 172938, 191919

Offset: 0

Author

Roger L. Bagula, Apr 11 2006

Keywords

• nonn
• easy

Comments

Also, the number of external intersections of the diagonals of a general n-gon = (A176145) - (A000332). - Michel Lagneau, Apr 21 2010

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..1000
• Aram Bingham, Lisa Johnston, Colin Lawson, Rosa Orellana, Jianping Pan, and Chelsea Sato, The Chromatic Symmetric Function for Unicyclic Graphs, arXiv:2505.06486 [math.CO], 2025. See p. 12.
• Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Crossrefs

Essentially the same as A050297 and A005701.

Cf. A000332, A176145.

Programs

Magma

[n*(n-1)*(n-2)*(n+3)/12: n in [0..50]]; // Vincenzo Librandi, Oct 10 2013

Maple

seq(n*(n-1)*(n-2)*(n+3)/12, n=0..40); # Wesley Ivan Hurt, Oct 10 2013

Mathematica

Table[n(n-1)(n-2)(n+3)/12, {n,0,100}] (* Wesley Ivan Hurt, Sep 26 2013 *)
CoefficientList[Series[x^3 (3 - x)/(1 - x)^5, {x, 0, 80}], x] (* Vincenzo Librandi, Oct 10 2013 *)
LinearRecurrence[{5,-10,10,-5,1},{0,0,0,3,14},80] (* Harvey P. Dale, Jan 01 2025 *)

Formula

G.f.: x^3*(3-x)/(1-x)^5. - Colin Barker, Jan 31 2012

From Amiram Eldar, May 17 2025: (Start)

Sum_{n>=3} 1/a(n) = 137/300.

Sum_{n>=3} (-1)^(n+1)/a(n) = 32*log(2)/5 - 1247/300. (End)

Extensions

Edited by N. J. A. Sloane, Apr 23 2006