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%I #37 Dec 22 2023 14:15:10
%S 1,1,1,1,2,2,1,3,4,2,1,4,8,6,3,1,5,14,18,9,3,1,6,22,44,39,12,4,1,7,32,
%T 90,135,81,16,4,1,8,44,162,363,408,166,20,5,1,9,58,266,813,1455,1228,
%U 336,25,5,1,10,74,408,1599,4068,5824,3688,677,30,6
%N Square array T(n,k), n >= 2, k >= 0, read by antidiagonals downwards, where T(n,k) = Sum_{j=0..n} k^(n-j) * floor(j/2).
%H Seiichi Manyama, <a href="/A368296/b368296.txt">Antidiagonals n = 2..141, flattened</a>
%F T(n,k) = T(n-2,k) + Sum_{j=0..n-2} k^j.
%F T(n,k) = 1/(k+1) * (-floor((n+1)/2) + Sum_{j=1..n} j*k^(n-j)).
%F T(n,k) = 1/(k-1) * Sum_{j=0..n} floor(k^j/(k+1)) = Sum_{j=0..n} floor(k^j/(k^2-1)) for k > 1.
%F T(n,k) = (k+1)*T(n-1,k) - (k-1)*T(n-2,k) - (k+1)*T(n-3,k) + k*T(n-4,k).
%F G.f. of column k: x^2/((1-x) * (1-k*x) * (1-x^2)).
%F T(n,k) = 1/(k-1) * (floor(k^(n+1)/(k^2-1)) - floor((n+1)/2)) for k > 1.
%e Square array begins:
%e 1, 1, 1, 1, 1, 1, 1, ...
%e 1, 2, 3, 4, 5, 6, 7, ...
%e 2, 4, 8, 14, 22, 32, 44, ...
%e 2, 6, 18, 44, 90, 162, 266, ...
%e 3, 9, 39, 135, 363, 813, 1599, ...
%e 3, 12, 81, 408, 1455, 4068, 9597, ...
%e 4, 16, 166, 1228, 5824, 20344, 57586, ...
%o (PARI) T(n, k) = (-((n+1)\2)+sum(j=1, n, j*k^(n-j)))/(k+1);
%Y Columns k=0..8 give A004526, A002620, A178420, A097137, A097138, A097139, A178719, A178730, A178827.
%Y Cf. A055129, A126885, A368343.
%K nonn,tabl,easy
%O 2,5
%A _Seiichi Manyama_, Dec 20 2023
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