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A165507 Triangle T(n,m) read by rows: numerator of 1/(1+n-m)^2 - 1/m^2.

%I A165507 #13 Sep 08 2022 08:45:47
%S A165507 0,-3,3,-8,0,8,-15,-5,5,15,-24,-3,0,3,24,-35,-21,-7,7,21,35,-48,-2,
%T A165507 -16,0,16,2,48,-63,-45,-1,-9,9,1,45,63,-80,-15,-40,-5,0,5,40,15,80,
%U A165507 -99,-77,-55,-33,-11,11,33,55,77,99,-120,-6,-8,-3,-24,0,24,3,8,6,120
%N A165507 Triangle T(n,m) read by rows: numerator of 1/(1+n-m)^2 - 1/m^2.
%C A165507 The triangle is obtained from the infinite array shown in the comment in A172370 by starting in column 1 and reading diagonally upwards along increasing columns or starting in column -1 and reading diagonally upwards along decreasing columns.
%C A165507 Equivalence of these two interpretations follows from the mirror symmetry m <-> -m along column m=0 in that array.
%C A165507 T(n,m) is antisymmetric (changes sign) with respect to a central zero if the row index n is odd, and with respect to the separator in the middle of the row if the row index n is even: T(n,m) = -T(n,n+1-m).
%C A165507 An appropriate triangle of denominators is in A143183.
%H A165507 G. C. Greubel, <a href="/A165507/b165507.txt">Rows n=1..100 of triangle, flattened</a>
%F A165507 T(n,m) = A173651(1+n,m), m>=1.
%e A165507 The triangle starts in row n=1 with columns 1<=m<=n as
%e A165507 0;
%e A165507 -3,3;
%e A165507 -8,0,8;
%e A165507 -15,-5,5,15;
%e A165507 -24,-3,0,3,24;
%e A165507 -35,-21,-7,7,21,35;
%e A165507 -48,-2,-16,0,16,2,48;
%p A165507 A165507 := proc(n,m) 1/(1+n-m)^2-1/m^2 ; numer(%) ; end proc:
%t A165507 Table[Numerator[1/(n-k+1)^2 - 1/k^2], {n,1,15}, {k,1,n}]//Flatten (* _G. C. Greubel_, Oct 21 2018 *)
%o A165507 (PARI) for(n=1, 15, for(k=1, n, print1(numerator(1/(n-k+1)^2 - 1/k^2), ", "))) \\ _G. C. Greubel_, Oct 21 2018
%o A165507 (Magma) [[Numerator(1/(n-k+1)^2 - 1/k^2): k in [1..n]]: n in [1..15]]; // _G. C. Greubel_, Oct 21 2018
%Y A165507 Cf. A120072, A143183, A165441.
%K A165507 sign,frac,tabl,easy
%O A165507 1,2
%A A165507 _Paul Curtz_, Sep 21 2009