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Differences of squares: T(n,n) = n^2, T(n,k) = T(n,k+1) - T(n-1,k), 0 <= k < n, triangle read by rows.
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%I #10 Jul 05 2018 04:59:29

%S 0,1,1,2,3,4,0,2,5,9,0,0,2,7,16,0,0,0,2,9,25,0,0,0,0,2,11,36,0,0,0,0,

%T 0,2,13,49,0,0,0,0,0,0,2,15,64,0,0,0,0,0,0,0,2,17,81,0,0,0,0,0,0,0,0,

%U 2,19,100,0,0,0,0,0,0,0,0,0,2,21,121,0,0,0,0,0,0,0,0,0,0,2,23,144

%N Differences of squares: T(n,n) = n^2, T(n,k) = T(n,k+1) - T(n-1,k), 0 <= k < n, triangle read by rows.

%C T(n,n) = A000290(n);

%C T(n,n-1) = A005408(n-1), n > 0;

%C T(n,n-2) = A007395(n-2), n > 1;

%C T(n,n-j) = A000004(n-j), 3 <= j <= n;

%C sum of n-th row = if n <= 1 then 2*n else (n+1)^2.

%H G. C. Greubel, <a href="/A162593/b162593.txt">Rows n=0..99 of triangle, flattened</a>

%e From _Jon E. Schoenfield_, Jul 04 2018: (Start)

%e Table begins

%e .

%e n\k| 0 1 2 3 4 5 6 7 8 9 10 11 12

%e ---+--------------------------------------------------

%e 0 | 0

%e 1 | 1 1

%e 2 | 2 3 4

%e 3 | 0 2 5 9

%e 4 | 0 0 2 7 16

%e 5 | 0 0 0 2 9 25

%e 6 | 0 0 0 0 2 11 36

%e 7 | 0 0 0 0 0 2 13 49

%e 8 | 0 0 0 0 0 0 2 15 64

%e 9 | 0 0 0 0 0 0 0 2 17 81

%e 10 | 0 0 0 0 0 0 0 0 2 19 100

%e 11 | 0 0 0 0 0 0 0 0 0 2 21 121

%e 12 | 0 0 0 0 0 0 0 0 0 0 2 23 144

%e ...

%e (End)

%t T[n_, n_] := n^2; T[n_, k_] := T[n, k] = T[n, k + 1] - T[n - 1, k]; Table[T[n, k], {n, 0, 15}, {k, 0, n}] // Flatten (* _G. C. Greubel_, Jul 04 2018 *)

%Y Cf. A162594 (differences of cubes).

%K nonn,tabl

%O 0,4

%A _Reinhard Zumkeller_, Jul 07 2009