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 A165795 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. 5
 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 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 0,8 COMMENTS 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. LINKS G. C. Greubel, Antidiagonals n = 0..50, flattened FORMULA A(n, k) = numerator(1/n^2 - 1/k^2) with A(0,k) = 1 and A(n,0) = -1 (array). A(n, 0) = -A158388(n). A(n, k) = A172157(n,k), n>=1. From G. C. Greubel, Mar 10 2022: (Start) 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). T(2*n+1, n) = -A005408(n). (End) EXAMPLE 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...0...3...8..15..24..35..48..63..80..99. A005563, A147998 -1..-3...0...5...3..21...2..45..15..77...6. A061037, A070262 -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; MATHEMATICA 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 *) PROG (Sage) def A165795(n, k): if (k==n): return 1 elif (k==0): return -1 else: return numerator(1/(n-k)^2 -1/k^2) flatten([[A165795(n, k) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Mar 10 2022 CROSSREFS Cf. A158388, A165507, A172157, A173651. Cf. A005408, A005563, A061037, A061039, A070262, A147998. Sequence in context: A196904 A197373 A197529 * A287290 A287981 A213660 Adjacent sequences: A165792 A165793 A165794 * A165796 A165797 A165798 KEYWORD frac,tabl,easy,sign AUTHOR Paul Curtz, Sep 27 2009 STATUS approved

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Last modified September 28 19:19 EDT 2023. Contains 365737 sequences. (Running on oeis4.)