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

%I #8 Apr 26 2022 04:22:16

%S 1,2,2,1,2,1,-8,0,0,-8,-31,-4,5,-4,-31,-74,-10,22,22,-10,-74,-143,-18,

%T 57,82,57,-18,-143,-244,-28,116,188,188,116,-28,-244,-383,-40,205,352,

%U 401,352,205,-40,-383,-566,-54,330,586,714,714,586,330,-54,-566

%N Triangle T(n, k) = 1 + (n + k - n*k)*(2*n - k - n*(n-k)), read by rows.

%H G. C. Greubel, <a href="/A172176/b172176.txt">Rows n = 0..50 of the triangle, flattened</a>

%F T(n, k) = 1 + (n-(n-1)*k)*(n-(n-1)*(n-k)).

%F T(n, n-k) = T(n, k).

%F T(n, 0) = 1 - A027620(n-3).

%F T(n, 1) = -A028552(n-3).

%F T(n, 2) = A033445(n-2).

%F Sum_{k=0..n} T(n, k) = (n+1)*(n^4 - 9*n^3 + 15*n^2 - n + 6)/6.

%e Triangle begins as:

%e 1;

%e 2, 2;

%e 1, 2, 1;

%e -8, 0, 0, -8;

%e -31, -4, 5, -4, -31;

%e -74, -10, 22, 22, -10, -74;

%e -143, -18, 57, 82, 57, -18, -143;

%e -244, -28, 116, 188, 188, 116, -28, -244;

%e -383, -40, 205, 352, 401, 352, 205, -40, -383;

%e -566, -54, 330, 586, 714, 714, 586, 330, -54, -566;

%e -799, -70, 497, 902, 1145, 1226, 1145, 902, 497, -70, -799;

%p A172176:= proc(n,m) 1+(n+m-n*m)*(2*n-m-n*(n-m)); end proc:

%p seq(seq(A172176(n,m), m=0..n), n=0..12);

%t T[n_, k_]= 1 + (n-(n-1)*k)*(n-(n-1)*(n-k));

%t Table[T[n, k], {n,0,12}, {k,0,n}]//Flatten

%o (Magma) [1 + (n-(n-1)*k)*(n-(n-1)*(n-k)): k in [0..n], n in [0..12]]; // _G. C. Greubel_, Apr 26 2022

%o (SageMath)

%o def A172176(n,k): return 1 + (n-(n-1)*k)*(n-(n-1)*(n-k))

%o flatten([[A172176(n,k) for k in (0..n)] for n in (0..12)]) # _G. C. Greubel_, Apr 26 2022

%Y Cf. A027620, A028552, A033445.

%K sign,tabl,easy

%O 0,2

%A _Roger L. Bagula_, Jan 28 2010