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A173261 Array T(n,k) read by antidiagonals: T(n,2k)=1, T(n,2k+1)=n, n>=2, k>=0. 1

%I #13 Dec 09 2021 01:04:35

%S 1,1,2,1,3,1,1,4,1,2,1,5,1,3,1,1,6,1,4,1,2,1,7,1,5,1,3,1,1,8,1,6,1,4,

%T 1,2,1,9,1,7,1,5,1,3,1,1,10,1,8,1,6,1,4,1,2,1,11,1,9,1,7,1,5,1,3,1,1,

%U 12,1,10,1,8,1,6,1,4,1,2,1,13,1,11,1,9,1,7,1,5,1,3,1,1,14,1,12,1,10,1,8,1,6,1,4,1,2

%N Array T(n,k) read by antidiagonals: T(n,2k)=1, T(n,2k+1)=n, n>=2, k>=0.

%C One may define another array B(n,0) = -1, B(n,k) = T(n,k-1) + 2*B(n,k-1), n>=2, which also starts in columns k>=0, as follows:

%C -1, -1, 0, 1, 4, 9, 20, 41, 84, 169, 340, 681, 1364 ...: A084639;

%C -1, -1, 1, 3, 9, 19, 41, 83, 169, 339, 681, 1363, 2729;

%C -1, -1, 2, 5, 14, 29, 62, 125, 254, 509, 1022, 2045, 4094;

%C -1, -1, 3, 7, 19, 39, 83, 167, 339, 679, 1363, 2727, 5459 ...: -A173114;

%C B(n,k) = (n-1)*A001045(k) - T(n,k).

%C First differences are B(n,k+1) - B(n,k) = (n-1)*A001045(k).

%H G. C. Greubel, <a href="/A173261/b173261.txt">Antidiagonals n = 0..50 of the array, flattened</a>

%F From _G. C. Greubel_, Dec 03 2021: (Start)

%F T(n, k) = (1/2)*((n+3) - (n+1)*(-1)^k).

%F Sum_{k=0..n} T(n-k, k) = A024206(n).

%F Sum_{k=0..floor((n+2)/2)} T(n-2*k+2, k) = (1/16)*(2*n^2 4*n -5*(1 +(-1)^n) + 4*sin(n*Pi/2)) (diagonal sums).

%F T(2*n-2, n) = A093178(n). (End)

%e The array T(n,k) starts in row n=2 with columns k>=0 as:

%e 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2 ... A000034;

%e 1, 3, 1, 3, 1, 3, 1, 3, 1, 3, 1, 3 ... A010684;

%e 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4 ... A010685;

%e 1, 5, 1, 5, 1, 5, 1, 5, 1, 5, 1, 5 ... A010686;

%e 1, 6, 1, 6, 1, 6, 1, 6, 1, 6, 1, 6 ... A010687;

%e 1, 7, 1, 7, 1, 7, 1, 7, 1, 7, 1, 7 ... A010688;

%e 1, 8, 1, 8, 1, 8, 1, 8, 1, 8, 1, 8 ... A010689;

%e 1, 9, 1, 9, 1, 9, 1, 9, 1, 9, 1, 9 ... A010690;

%e 1, 10, 1, 10, 1, 10, 1, 10, 1, 10, 1, 10 ... A010691.

%e Antidiagonal triangle begins as:

%e 1;

%e 1, 2;

%e 1, 3, 1;

%e 1, 4, 1, 2;

%e 1, 5, 1, 3, 1;

%e 1, 6, 1, 4, 1, 2;

%e 1, 7, 1, 5, 1, 3, 1;

%e 1, 8, 1, 6, 1, 4, 1, 2;

%e 1, 9, 1, 7, 1, 5, 1, 3, 1;

%e 1, 10, 1, 8, 1, 6, 1, 4, 1, 2;

%e 1, 11, 1, 9, 1, 7, 1, 5, 1, 3, 1;

%e 1, 12, 1, 10, 1, 8, 1, 6, 1, 4, 1, 2;

%e 1, 13, 1, 11, 1, 9, 1, 7, 1, 5, 1, 3, 1;

%e 1, 14, 1, 12, 1, 10, 1, 8, 1, 6, 1, 4, 1, 2;

%t T[n_, k_]:= (1/2)*((n+3) - (n+1)*(-1)^k);

%t Table[T[n-k, k], {n,2,17}, {k,2,n}]//Flatten (* _G. C. Greubel_, Dec 03 2021 *)

%o (Sage) flatten([[(1/2)*((n-k+3) - (n-k+1)*(-1)^k) for k in (2..n)] for n in (2..17)]) # _G. C. Greubel_, Dec 03 2021

%Y Cf. A000034, A010684, A010685, A010686, A010687, A010688, A010689, A010690, A010691.

%Y Cf. A001045, A024206, A084639, A093178, A173114.

%K nonn,tabl,easy

%O 2,3

%A _Paul Curtz_, Feb 14 2010

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