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A257238 Triangle T(n, k) = n^3 - k^3, 0 <= k < = n.

%I A257238 #23 Apr 05 2025 03:06:09
%S A257238 0,1,0,8,7,0,27,26,19,0,64,63,56,37,0,125,124,117,98,61,0,216,215,208,
%T A257238 189,152,91,0,343,342,335,316,279,218,127,0,512,511,504,485,448,387,
%U A257238 296,169,0,729,728,721,702,665,604,513,386,217,0,1000,999,992,973,936,875,784,657,488,271,0
%N A257238 Triangle T(n, k) = n^3 - k^3, 0 <= k < = n.
%C A257238 See the comments in A025581 and A079904 on a problem by François Viète (Vieta)(1593). Regarding that problem, note the simple identity: n^3 - k^3 = (n - k)^3 + 3*n*k*(n - k), for n > = k >= 0.
%C A257238 Row sums give A126274(n-1) for n >= 1, and 0 for n=0.
%C A257238 Alternating row sums are ars(2*n) = ars(2*n-1) = (4*n-3)*n^2 = A103532(n-1), for n >= 1, and ars(0) = 0.
%H A257238 Robert Israel, <a href="/A257238/b257238.txt">Table of n, a(n) for n = 0..10010</a> (rows 0 to 140, flattened)
%F A257238 T(n, k) =  A025581(n, k)*(A025581(n, k)^2 + 3* A079904(n, k)) (see the identity mentioned in a comment).
%F A257238 Columns (with one leading zero and offset 0): k=0: l^3 = A000578(l), k=1: (l+1)^3 - 1 = A068601(l+1), k=2: l*(l^2 + 6*l + 12), k=3: l*(l^2 + 9*l + 27), k=4: l*(l^2 + 12*l + 48), k=5: l*(l^2 + 15*l + 75), ...
%F A257238 G.f. for T(n,k): (1+4*x+4*x*y+x^2-14*x^2*y+x^2*y^2-2*x^3*y-2*x^3*y^2+7*x^4*y^2)*x/((1-x*y)^3*(1-x)^4). - _Robert Israel_, May 10 2018
%e A257238 The triangle T(n, k) begins:
%e A257238   n\k    0   1   2   3   4   5   6   7   8   9  10
%e A257238    0:    0
%e A257238    1:    1   0
%e A257238    2:    8   7   0
%e A257238    3:   27  26  19   0
%e A257238    4:   64  63  56  37   0
%e A257238    5:  125 124 117  98  61   0
%e A257238    6:  216 215 208 189 152  91   0
%e A257238    7:  343 342 335 316 279 218 127   0
%e A257238    8:  512 511 504 485 448 387 296 169   0
%e A257238    9:  729 728 721 702 665 604 513 386 217   0
%e A257238   10: 1000 999 992 973 936 875 784 657 488 271   0
%e A257238   ...
%p A257238 for n from 0 to 10 do seq(n^3-k^3,k=0..n) od; # _Robert Israel_, May 10 2018
%t A257238 Table[n^3-k^3,{n,0,10},{k,0,n}]//Flatten (* _Harvey P. Dale_, Jan 02 2021 *)
%Y A257238 Cf. A126274, A103532, A025581, A079904, A335821.
%K A257238 nonn,easy,tabl
%O A257238 0,4
%A A257238 _Wolfdieter Lang_, May 12 2015