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%I #12 Mar 11 2022 08:31:03
%S 1,-1,1,-1,0,1,-1,-3,3,1,-1,-8,0,8,1,-1,-15,-5,5,15,1,-1,-24,-3,0,3,
%T 24,1,-1,-35,-21,-7,7,21,35,1,-1,-48,-2,-16,0,16,2,48,1,-1,-63,-45,-1,
%U -9,9,1,45,63,1,-1,-80,-15,-40,-5,0,5,40,15,80,1,-1,-99,-77,-55,-33,-11,11,33,55,77,99,1
%N Array A(n, k) = numerator of 1/n^2 - 1/k^2 with A(0,k) = 1 and A(n,0) = -1, read by antidiagonals.
%C A row of A(0,k)= 1 is added on top of the array shown in A172157, which is then read upwards by antidiagonals.
%C One may also interpret this as appending a 1 to each row of A173651 or adding a column of -1's and a diagonal of +1's to A165507.
%H G. C. Greubel, <a href="/A165795/b165795.txt">Antidiagonals n = 0..50, flattened</a>
%F A(n, k) = numerator(1/n^2 - 1/k^2) with A(0,k) = 1 and A(n,0) = -1 (array).
%F A(n, 0) = -A158388(n).
%F A(n, k) = A172157(n,k), n>=1.
%F From _G. C. Greubel_, Mar 10 2022: (Start)
%F T(n, k) = numerator(1/(n-k)^2 -1/k^2), with T(n,n) = 1, T(n,0) = -1 (triangle).
%F A(n, n) = T(2*n, n) = 0^n.
%F Sum_{k=0..n} T(n, k) = 0^n.
%F T(n, n-k) = -T(n,k).
%F T(2*n+1, n) = -A005408(n). (End)
%e The array, A(n, k), of numerators starts in row n=0 with columns m>=0 as:
%e .1...1...1...1...1...1...1...1...1...1...1.
%e -1...0...3...8..15..24..35..48..63..80..99. A005563, A147998
%e -1..-3...0...5...3..21...2..45..15..77...6. A061037, A070262
%e -1..-8..-5...0...7..16...1..40..55...8..91. A061039
%e Antidiagonal triangle, T(n, k), begins as:
%e 1;
%e -1, 1;
%e -1, 0, 1;
%e -1, -3, 3, 1;
%e -1, -8, 0, 8, 1;
%e -1, -15, -5, 5, 15, 1;
%e -1, -24, -3, 0, 3, 24, 1;
%e -1, -35, -21, -7, 7, 21, 35, 1;
%t T[n_, k_]:= If[k==n, 1, If[k==0, -1, Numerator[1/(n-k)^2 - 1/k^2]]];
%t Table[T[n, k], {n,0,12}, {k,0,n}]//Flatten (* _G. C. Greubel_, Mar 10 2022 *)
%o (Sage)
%o def A165795(n,k):
%o if (k==n): return 1
%o elif (k==0): return -1
%o else: return numerator(1/(n-k)^2 -1/k^2)
%o flatten([[A165795(n,k) for k in (0..n)] for n in (0..12)]) # _G. C. Greubel_, Mar 10 2022
%Y Cf. A158388, A165507, A172157, A173651.
%Y Cf. A005408, A005563, A061037, A061039, A070262, A147998.
%K frac,tabl,easy,sign
%O 0,8
%A _Paul Curtz_, Sep 27 2009
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