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A symmetrical triangle sequence: T(n, k) = q^k + q^(n-k) - q^n, with q=2.
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%I #7 Sep 08 2022 08:45:52

%S 1,1,1,1,0,1,1,-2,-2,1,1,-6,-8,-6,1,1,-14,-20,-20,-14,1,1,-30,-44,-48,

%T -44,-30,1,1,-62,-92,-104,-104,-92,-62,1,1,-126,-188,-216,-224,-216,

%U -188,-126,1,1,-254,-380,-440,-464,-464,-440,-380,-254,1

%N A symmetrical triangle sequence: T(n, k) = q^k + q^(n-k) - q^n, with q=2.

%C Row sums are: {1, 2, 2, -2, -18, -66, -194, -514, -1282, -3074, -7170, ...}.

%H G. C. Greubel, <a href="/A176224/b176224.txt">Rows n = 0..100 of triangle, flattened</a>

%F T(n,k) = q^k + q^(n-k) - q^n, with q = 2.

%e Triangle begins as:

%e 1;

%e 1, 1;

%e 1, 0, 1;

%e 1, -2, -2, 1;

%e 1, -6, -8, -6, 1;

%e 1, -14, -20, -20, -14, 1;

%e 1, -30, -44, -48, -44, -30, 1;

%e 1, -62, -92, -104, -104, -92, -62, 1;

%e 1, -126, -188, -216, -224, -216, -188, -126, 1;

%e 1, -254, -380, -440, -464, -464, -440, -380, -254, 1;

%e 1, -510, -764, -888, -944, -960, -944, -888, -764, -510, 1;

%p q:=2; seq(seq(q^k +q^(n-k) -q^n, k=0..n), n=0..12); # _G. C. Greubel_, Nov 23 2019

%t T[n_, k_, q_]:= q^k +q^(n-k) -q^n; Table[T[n, k, 2], {n, 0, 12}, {k, 0, n}]//Flatten (* modified by _G. C. Greubel_, Nov 23 2019 *)

%o (PARI) T(n,k,q) = my(q=2); q^k +q^(n-k) -q^n; \\ _G. C. Greubel_, Nov 23 2019

%o (Magma) q:=2; [q^k +q^(n-k) -q^n : k in [0..n], n in [0..12]]; // _G. C. Greubel_, Nov 23 2019

%o (Sage) q=2; [[q^k +q^(n-k) -q^n for k in (0..n)] for n in (0..12)] # _G. C. Greubel_, Nov 23 2019

%o (GAP) q:=2;; Flat(List([0..12], n-> List([0..n], k-> q^k +q^(n-k) -q^n ))); # _G. C. Greubel_, Nov 23 2019

%Y Cf. this sequence (q=2), A176225 (q=3), A176226 (q=5), A176227 (q=4).

%K sign,tabl

%O 0,8

%A _Roger L. Bagula_, Apr 12 2010

%E Edited by _G. C. Greubel_, Nov 23 2019