%I #15 Apr 23 2024 08:28:13
%S -1,3,3,19,4,19,53,17,17,53,111,48,27,48,111,199,103,55,55,103,199,
%T 323,188,107,80,107,188,323,489,309,189,129,129,189,309,489,703,472,
%U 307,208,175,208,307,472,703,971,683,467,323,251,251,323,467,683,971,1299,948,675,480,363,324,363,480,675,948,1299
%N Triangle read by rows: T(n, k) = (n-k+1)^3 + (k+1)^3 - 3*(n-k+1)*(k+1).
%H G. C. Greubel, <a href="/A143180/b143180.txt">Rows n = 0..50 of the triangle, flattened</a>
%F T(n, k) = (n-k+1)^3 + (k+1)^3 - 3*(n-k+1)*(k+1).
%F T(n, n-k) = T(n, k).
%F G.f.: ((-1 + 7*x + x^2 - x^3) + (7 - 36*x + 15*x^2 - 4*x^3)*(x*y) + (1 + 15*x + 3*x^2 - x^3)*(x*y)^2 - (1 + 4*x + x^2)*(x*y)^3)/((1-x)^4*(1 - x*y)^4). - _G. C. Greubel_, Apr 22 2024
%e Triangle begins as:
%e -1;
%e 3, 3;
%e 19, 4, 19;
%e 53, 17, 17, 53;
%e 111, 48, 27, 48, 111;
%e 199, 103, 55, 55, 103, 199;
%e 323, 188, 107, 80, 107, 188, 323;
%e 489, 309, 189, 129, 129, 189, 309, 489;
%e 703, 472, 307, 208, 175, 208, 307, 472, 703;
%e 971, 683, 467, 323, 251, 251, 323, 467, 683, 971;
%e 1299, 948, 675, 480, 363, 324, 363, 480, 675, 948, 1299;
%t T[n_,k_]:= (n-k+1)^3 + (k+1)^3 - 3*(n-k+1)*(k+1);
%t Table[T[n,k], {n,0,12}, {k,0,n}]//Flatten
%o (Magma)
%o A143180:= func< n,k | (n-k+1)^3 + (k+1)^3 - 3*(n-k+1)*(k+1) >;
%o [A143180(n,k): k in [0..n], n in [0..12]]; // _G. C. Greubel_, Apr 19 2024
%o (SageMath)
%o def A143180(n,k): return (n-k+1)^3 + (k+1)^3 - 3*(n-k+1)*(k+1)
%o flatten([[A143180(n,k) for k in range(n+1)] for n in range(13)]) # _G. C. Greubel_, Apr 19 2024
%Y Cf. A295709.
%K sign,tabl
%O 0,2
%A _Roger L. Bagula_ and _Gary W. Adamson_, Oct 17 2008
%E Edited by _G. C. Greubel_, Apr 19 2024