A165795 - OEIS

%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