A165507
Triangle T(n,m) read by rows: numerator of 1/(1+n-m)^2 - 1/m^2.
Original entry on oeis.org
0, -3, 3, -8, 0, 8, -15, -5, 5, 15, -24, -3, 0, 3, 24, -35, -21, -7, 7, 21, 35, -48, -2, -16, 0, 16, 2, 48, -63, -45, -1, -9, 9, 1, 45, 63, -80, -15, -40, -5, 0, 5, 40, 15, 80, -99, -77, -55, -33, -11, 11, 33, 55, 77, 99, -120, -6, -8, -3, -24, 0, 24, 3, 8, 6, 120
Offset: 1
The triangle starts in row n=1 with columns 1<=m<=n as
0;
-3,3;
-8,0,8;
-15,-5,5,15;
-24,-3,0,3,24;
-35,-21,-7,7,21,35;
-48,-2,-16,0,16,2,48;
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[[Numerator(1/(n-k+1)^2 - 1/k^2): k in [1..n]]: n in [1..15]]; // G. C. Greubel, Oct 21 2018
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A165507 := proc(n,m) 1/(1+n-m)^2-1/m^2 ; numer(%) ; end proc:
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Table[Numerator[1/(n-k+1)^2 - 1/k^2], {n,1,15}, {k,1,n}]//Flatten (* G. C. Greubel, Oct 21 2018 *)
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for(n=1, 15, for(k=1, n, print1(numerator(1/(n-k+1)^2 - 1/k^2), ", "))) \\ G. C. Greubel, Oct 21 2018
A165727
Table T(k,n) read by antidiagonals: denominator of 1/min(n,k)^2 -1/max(n,k)^2 with T(0,n) = T(k,0) = 0.
Original entry on oeis.org
0, 0, 0, 0, 1, 0, 0, 4, 4, 0, 0, 9, 1, 9, 0, 0, 16, 36, 36, 16, 0, 0, 25, 16, 1, 16, 25, 0, 0, 36, 100, 144, 144, 100, 36, 0, 0, 49, 9, 225, 1, 225, 9, 49, 0, 0, 64, 196, 12, 400, 400, 12, 196, 64, 0, 0, 81, 64, 441, 144, 1, 144, 441, 64, 81, 0, 0, 100, 324, 576, 784, 900, 900, 784, 576, 324, 100, 0
Offset: 0
0, 0, 0, 0, 0, 0, 0, 0, 0, 0, ... A000004
0, 1, 4, 9, 16, 25, 36, 49, 64, 81, ... A000290
0, 4, 1, 36, 16, 100, 9, 196, 64, 324, ... A061038
0, 9, 36, 1, 144, 225, 12, 441, 576, 81, ... A061040
0, 16, 16, 144, 1, 400, 144, 784, 64, 1296, ... A061042
0, 25, 100, 225, 400, 1, 900, 1225, 1600, 2025, ... A061044
0, 36, 9, 12, 144, 900, 1, 1764, 576, 324, ... A061046
0, 49, 196, 441, 784, 1225, 1764, 1, 3136, 3969, ... A061048
0, 64, 64, 576, 64, 1600, 576, 3136, 1, 5184, ... A061050
0, 81, 324, 81, 1296, 2025, 324, 3969, 5184, 1, ...
Cf.
A165441 (top row and left column removed)
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T:= (k,n)-> `if` (n=0 or k=0, 0, denom (1/min (n,k)^2 -1/max (n, k)^2)):
seq (seq (T (k, d-k), k=0..d), d=0..11);
A174413
Triangle T(n,m) with the denominator of 1/(n-m)^2-1/n^2, read by rows, 1<=m
Original entry on oeis.org
4, 36, 9, 144, 16, 16, 400, 225, 100, 25, 900, 144, 12, 9, 36, 1764, 1225, 784, 441, 196, 49, 3136, 576, 1600, 64, 576, 64, 64, 5184, 3969, 324, 2025, 1296, 81, 324, 81, 8100, 1600, 4900, 225, 100, 400, 900, 25, 100, 12100, 9801, 7744, 5929, 4356, 3025, 1936, 1089, 484, 121
Offset: 2
Triangle T(n,m) begins:
4,
36, 9,
144, 16, 16,
400, 225, 100, 25,
900, 144, 12, 9, 36,
1764, 1225, 784, 441, 196, 49,
3136, 576, 1600, 64, 576, 64, 64,
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A174413 := proc(n,m) 1/(n-m)^2-1/n^2 ; denom(%) ; end proc:
seq(seq(A174413(n, k), k=1..n-1), n=2..11); # R. J. Mathar, Jan 27 2011
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T[n_, m_] := Denominator[1/(n - m)^2 - 1/n^2];
Table[T[n, m], {n, 2, 11}, {m, 1, n-1}] // Flatten (* Jean-François Alcover, May 18 2018 *)
Showing 1-3 of 3 results.
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