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%I #18 Jun 06 2021 08:59:25
%S 1,1,1,1,2,2,1,3,3,2,1,4,6,4,3,1,5,11,12,5,3,1,6,18,32,21,6,4,1,7,27,
%T 70,87,41,7,4,1,8,38,132,263,258,74,8,5,1,9,51,224,633,1047,745,144,9,
%U 5,1,10,66,352,1305,3158,4120,2224,275,10,6,1,11,83,522,2411,7821,15659,16460,6605,541,11,6
%N Square array T(n,k), n >= 1, k >= 0, read by antidiagonals downwards, where T(n,k) = Sum_{j=1..n} k^(floor(n/j) - 1).
%F G.f. of column k: (1/(1 - x)) * Sum_{j>=1} x^j * (1 - x^j)/(1 - k*x^j).
%e Square array begins:
%e 1, 1, 1, 1, 1, 1, 1, ...
%e 1, 2, 3, 4, 5, 6, 7, ...
%e 2, 3, 6, 11, 18, 27, 38, ...
%e 2, 4, 12, 32, 70, 132, 224, ...
%e 3, 5, 21, 87, 263, 633, 1305, ...
%e 3, 6, 41, 258, 1047, 3158, 7821, ...
%t T[n_, 0] := Floor[(n + 1)/2]; T[n_, k_] := Sum[k^(Floor[n/j] - 1), {j, 1, n}]; Table[T[k, n - k], {n, 1, 12}, {k, 1, n}] // Flatten (* _Amiram Eldar_, Jun 06 2021 *)
%o (PARI) T(n, k) = sum(j=1, n, k^(n\j-1));
%Y Columns k=0..3 give A110654, A000027, A345028, A345029.
%Y T(n,n) gives A345030.
%Y Cf. A344821, A345033.
%K nonn,tabl
%O 1,5
%A _Seiichi Manyama_, Jun 06 2021
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