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A165795
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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.
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5
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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, 24, 1, -1, -35, -21, -7, 7, 21, 35, 1, -1, -48, -2, -16, 0, 16, 2, 48, 1, -1, -63, -45, -1, -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
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OFFSET
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0,8
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COMMENTS
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A row of A(0,k)= 1 is added on top of the array shown in A172157, which is then read upwards by antidiagonals.
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.
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LINKS
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FORMULA
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A(n, k) = numerator(1/n^2 - 1/k^2) with A(0,k) = 1 and A(n,0) = -1 (array).
T(n, k) = numerator(1/(n-k)^2 -1/k^2), with T(n,n) = 1, T(n,0) = -1 (triangle).
A(n, n) = T(2*n, n) = 0^n.
Sum_{k=0..n} T(n, k) = 0^n.
T(n, n-k) = -T(n,k).
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EXAMPLE
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The array, A(n, k), of numerators starts in row n=0 with columns m>=0 as:
.1...1...1...1...1...1...1...1...1...1...1.
-1..-8..-5...0...7..16...1..40..55...8..91. A061039
Antidiagonal triangle, T(n, k), begins as:
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, 24, 1;
-1, -35, -21, -7, 7, 21, 35, 1;
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MATHEMATICA
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T[n_, k_]:= If[k==n, 1, If[k==0, -1, Numerator[1/(n-k)^2 - 1/k^2]]];
Table[T[n, k], {n, 0, 12}, {k, 0, n}]//Flatten (* G. C. Greubel, Mar 10 2022 *)
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PROG
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(Sage)
if (k==n): return 1
elif (k==0): return -1
else: return numerator(1/(n-k)^2 -1/k^2)
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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