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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, 212420, 234520

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OFFSET

0,4

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).

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)

From Elmo R. Oliveira, Nov 30 2025: (Start)

E.g.f.: x^3*exp(x)*(x + 6)/12.

a(n) = 5*a(n-1) - 10*a(n-2) + 10*a(n-3) - 5*a(n-4) + a(n-5). (End)

MAPLE

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

MATHEMATICA