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Triangle read by rows: T(n, k) = (n-k+1)^3 + (k+1)^3 - 3*(n-k+1)*(k+1).
2

%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