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A134449 Sum of even products minus sum of odd products of different pairs of numbers from 1 to n.

%I A134449 #35 Dec 09 2024 15:54:02
%S A134449 0,2,5,29,39,129,150,374,410,860,915,1707,1785,3059,3164,5084,5220,
%T A134449 7974,8145,11945,12155,17237,17490,24114,24414,32864,33215,43799,
%U A134449 44205,57255,57720,73592,74120,93194,93789,116469,117135,143849,144590,175790
%N A134449 Sum of even products minus sum of odd products of different pairs of numbers from 1 to n.
%H A134449 Colin Barker, <a href="/A134449/b134449.txt">Table of n, a(n) for n = 1..500</a>
%H A134449 <a href="/index/Rec#order_09">Index entries for linear recurrences with constant coefficients</a>, signature (1,4,-4,-6,6,4,-4,-1,1).
%F A134449 Empirical g.f.: x^2*(x^5-6*x^4+2*x^3-16*x^2-3*x-2) / ((x-1)^5*(x+1)^4). - _Colin Barker_, Sep 03 2013
%F A134449 Conjectures from _Colin Barker_, Mar 20 2015: (Start)
%F A134449 a(n) = (n^4+4*n^3-2*n^2-4*n)/16 for n even.
%F A134449 a(n) = (n^4-1)/16 for n odd. (End)
%F A134449 The above conjectures are true. - _Sela Fried_, Dec 08 2024
%F A134449 E.g.f.: (x*(1 + 17*x + 6*x^2 + x^3)*cosh(x) - (1 + x - 7*x^2 - 10*x^3 - x^4)*sinh(x))/16. - _Stefano Spezia_, Dec 09 2024
%e A134449 {1,2,3} -> 1*2-1*3+2*3 = 5.
%e A134449 {1,2,3,4} -> 1*2-1*3+1*4+2*3+2*4+3*4 = 29.
%e A134449 {1,2,3,4,5} -> 1*2-1*3+1*4-1*5+2*3+2*4+2*5+3*4-3*5+4*5 = 39.
%p A134449 P:=proc(n) local a,i,j,k,w; for i from 1 by 1 to n do a:=0; for j from 1 by 1 to i do w:=j; k:=i; while k>w do a:=a+w*k*(-1)^(w*k); k:=k-1; od; od; print(a); od; end: P(100);
%t A134449 epop[n_]:=Module[{f=Times@@@Subsets[n,{2}]},Total[Select[f,EvenQ]]-Total[ Select[ f,OddQ]]]; Table[epop[Range[n]],{n,40}] (* _Harvey P. Dale_, Sep 17 2017 *)
%o A134449 (PARI) a(n) = {s = 0; for (i=1, n, for (j=i+1, n, p = i*j; if (p % 2, s -= p, s += p););); s;} \\ _Michel Marcus_, Mar 20 2015
%Y A134449 Cf. A000914, A000915.
%K A134449 nonn,easy
%O A134449 1,2
%A A134449 _Paolo P. Lava_ and _Giorgio Balzarotti_, Jan 31 2008