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%I #21 Jul 28 2023 16:25:03
%S 3,5,8,7,3,15,9,16,21,24,11,5,1,2,35,13,24,33,40,45,48,15,7,39,3,55,
%T 15,63,17,32,5,56,65,8,77,80,19,9,51,4,3,21,91,6,99,21,40,57,72,85,96,
%U 105,112,117,120,23,11,7,5,95,1,119,1,5,35,143,25,48,69,88,105,120,133,144
%N Mirrored triangle A120072 read by rows.
%C A table of numerators of 1/n^2 - 1/m^2 extended to negative m looks as follows, stacked such that values of common m are aligned
%C and the central column of -1 is defined for m=0:
%C .............................0..-1...0...3...8..15..24..35..48..63..80..99. A005563
%C .........................0..-3..-1..-3...0...5...3..21...2..45..15..77...6. A061037
%C .....................0..-5..-8..-1..-8..-5...0...7..16...1..40..55...8..91. A061039
%C .................0..-7..-3.-15..-1.-15..-3..-7...0...9...5..33...3..65..21. A061041
%C .............0..-9.-16.-21.-24..-1.-24.-21.-16..-9...0..11..24..39..56...3. A061043
%C .........0.-11..-5..-1..-2.-35..-1.-35..-2..-1..-5.-11...0..13...7...5...4. A061045
%C .....0.-13.-24.-33.-40.-45.-48..-1.-48.-45.-40.-33.-24.-13...0..15..32..51. A061047
%C .0.-15..-7.-39..-3.-55.-15.-63..-1.-63.-15.-55..-3.-39..-7.-15...0..17...9. A061049
%C The row-reversed variant of A120072 appears (negated) after the leftmost 0.
%C Equals A061035 with the first column removed. - _Georg Fischer_, Jul 26 2023
%H G. C. Greubel, <a href="/A172370/b172370.txt">Rows n = 2..100 of triangle, flattened</a>
%F T(n,m) = numerator of 1/(n-m)^2 - 1/n^2, n >= 2, 1 <= m < n. - _R. J. Mathar_, Nov 23 2010
%e The table starts
%e 3
%e 5 8
%e 7 3 15
%e 9 16 21 24
%e 11 5 1 2 35
%e 13 24 33 40 45 48
%e 15 7 39 3 55 15 63
%e 17 32 5 56 65 8 77 80
%e 19 9 51 4 3 21 91 6 99
%t Table[Numerator[1/(n-k)^2 -1/n^2], {n, 2, 20}, {k, 1, n-1}]//Flatten (* _G. C. Greubel_, Sep 20 2018 *)
%o (PARI) for(n=2,20, for(k=1,n-1, print1(numerator(1/(n-k)^2 -1/n^2), ", "))) \\ _G. C. Greubel_, Sep 20 2018
%o (Magma) [[Numerator(1/(n-k)^2 -1/n^2): k in [1..n-1]]: n in [2..20]]; // _G. C. Greubel_, Sep 20 2018
%Y Lower diagonal gives: A070262, A061037(n+2).
%Y Cf. A061035, A172157, A165795.
%K nonn,easy,tabl
%O 2,1
%A _Paul Curtz_, Feb 01 2010
%E Comment rewritten and offset set to 2 by _R. J. Mathar_, Nov 23 2010