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

0, 2, 5, 29, 39, 129, 150, 374, 410, 860, 915, 1707, 1785, 3059, 3164, 5084, 5220, 7974, 8145, 11945, 12155, 17237, 17490, 24114, 24414, 32864, 33215, 43799, 44205, 57255, 57720, 73592, 74120, 93194, 93789, 116469, 117135, 143849, 144590, 175790

Offset

1,2

Links

Colin Barker, Table of n, a(n) for n = 1..500

Index entries for linear recurrences with constant coefficients, signature (1,4,-4,-6,6,4,-4,-1,1).

Formula

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

Conjectures from Colin Barker, Mar 20 2015: (Start)

a(n) = (n^4+4*n^3-2*n^2-4*n)/16 for n even.

a(n) = (n^4-1)/16 for n odd. (End)

The above conjectures are true. - Sela Fried, Dec 08 2024

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

Example

{1,2,3} -> 1*2-1*3+2*3 = 5.
{1,2,3,4} -> 1*2-1*3+1*4+2*3+2*4+3*4 = 29.
{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.

Maple

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);

Mathematica

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 *)

Prog

(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

Crossrefs

Cf. A000914, A000915.

Sequence in context: A098858 A213995 A370513 * A103579 A161500 A061351

Adjacent sequences: A134446 A134447 A134448 * A134450 A134451 A134452

Keyword

nonn,easy

Author

Paolo P. Lava and Giorgio Balzarotti, Jan 31 2008

Status

approved