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Square array T(n,k) read by antidiagonals upwards: T(n,0)=1; T(n,1) = n+1; T(n,2) = 2n+1, T(n,k>2) = T(n,k-1) - T(n,k-2) - T(n,k-3).
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%I #45 May 27 2020 20:52:42

%S 1,1,1,1,2,1,1,3,3,-1,1,4,5,0,-3,1,5,7,1,-5,-3,1,6,9,2,-7,-8,1,1,7,11,

%T 3,-9,-13,-3,7,1,8,13,4,-11,-18,-7,10,9,1,9,15,5,-13,-23,-11,13,21,1,

%U 1,10,17,6,-15,-28,-15,16,33,14,-15

%N Square array T(n,k) read by antidiagonals upwards: T(n,0)=1; T(n,1) = n+1; T(n,2) = 2n+1, T(n,k>2) = T(n,k-1) - T(n,k-2) - T(n,k-3).

%F T(0,k) = A180735(k-1).

%F T(n,k) - T(n-1,k) = -A078016(k+1).

%e Array starts:

%e 1 1 1 -1 -3 -3 1 7 9 1 -15 -25

%e 1 2 3 0 -5 -8 -3 10 21 14 -17 -52

%e 1 3 5 1 -7 -13 -7 13 33 27 -19 -79

%e 1 4 7 2 -9 -18 -11 16 45 40 -21 -106

%e 1 5 9 3 -11 -23 -15 19 57 53 -23 -133

%e 1 6 11 4 -13 -28 -19 22 69 66 -25 -160

%e 1 7 13 5 -15 -33 -23 25 81 79 -27 -187

%t T[n_, k_]:= T[n, k]= If[k<3, k*n+1, T[n, k-1] - T[n, k-2] - T[n, k-3]];

%t Table[T[n-k, k], {n,0,12}, {k,0,n}]//Flatten (* _G. C. Greubel_, May 26 2020 *)

%Y Cf. A078016, A180735.

%Y Columns k: A000012 (k=0), A000027 (k=1), A005408 (k=2), A023443 (k=3), A165747 (k=4), -A016885 (k=5), -A004767 (k=6), A016777 (k=7), A017629 (k=8), A190991 (k=9).

%K sign,tabl

%O 0,5

%A _Bob Selcoe_, May 05 2020