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A337130 a(n) is the sum of all products of pairs of numbers joined by the diagonals of an n-gon when its vertices are numbered from 1 to n in order.

%I A337130 #45 Sep 03 2020 12:25:13
%S A337130 0,0,0,11,40,99,203,370,621,980,1474,2133,2990,4081,5445,7124,9163,
%T A337130 11610,14516,17935,21924,26543,31855,37926,44825,52624,61398,71225,
%U A337130 82186,94365,107849,122728,139095,157046,176680,198099,221408,246715,274131,303770
%N A337130 a(n) is the sum of all products of pairs of numbers joined by the diagonals of an n-gon when its vertices are numbered from 1 to n in order.
%C A337130 For n < 4, no n-gon has a diagonal and thus a(n)=0.
%H A337130 <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (5,-10,10,-5,1).
%F A337130 a(n) = 3*binomial(n+1, 4) - n = (n-2)*(n-1)*n*(n+1)/8 - n for n>=3; a(1) = a(2) = 0.
%F A337130 a(n) = A000914(n-1) - A006527(n).
%F A337130 From _Colin Barker_, Aug 19 2020: (Start)
%F A337130 G.f.: x^4*(11 - 15*x + 9*x^2 - 2*x^3) / (1 - x)^5.
%F A337130 a(n) = 5*a(n-1) - 10*a(n-2) + 10*a(n-3) - 5*a(n-4) + a(n-5) for n>7.
%F A337130 (End)
%F A337130 E.g.f.: x + x^2 + exp(x)*x*(-8 + 4*x^2 + x^3)/8. - _Stefano Spezia_, Aug 19 2020
%e A337130 The diagonals of 4-gon would be numbered (1,3) and (2,4). So a(4) = 1*3 + 2*4 = 11.
%e A337130 The diagonals of 5-gon would be numbered (1,3), (1,4), (2,4), (2,5) and (3,5). So a(5) = 1*3 + 1*4 + 2*4 + 2*5 + 3*5 = 40.
%o A337130 (PARI) concat([0,0,0],Vec(x^4*(11 - 15*x + 9*x^2 - 2*x^3) / (1 - x)^5 + O(x^40))) \\ _Colin Barker_, Aug 19 2020
%Y A337130 Partial sums of A117560. Cf. A000914 (products including sides), A007569, A007678.
%K A337130 nonn,easy
%O A337130 1,4
%A A337130 _Mohammed Yaseen_, Aug 17 2020