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A345032 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).

%I A345032 #18 Jun 06 2021 08:59:25
%S A345032 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 A345032 70,87,41,7,4,1,8,38,132,263,258,74,8,5,1,9,51,224,633,1047,745,144,9,
%U A345032 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 A345032 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 A345032 G.f. of column k: (1/(1 - x)) * Sum_{j>=1} x^j * (1 - x^j)/(1 - k*x^j).
%e A345032 Square array begins:
%e A345032   1, 1,  1,   1,    1,    1,    1, ...
%e A345032   1, 2,  3,   4,    5,    6,    7, ...
%e A345032   2, 3,  6,  11,   18,   27,   38, ...
%e A345032   2, 4, 12,  32,   70,  132,  224, ...
%e A345032   3, 5, 21,  87,  263,  633, 1305, ...
%e A345032   3, 6, 41, 258, 1047, 3158, 7821, ...
%t A345032 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 A345032 (PARI) T(n, k) = sum(j=1, n, k^(n\j-1));
%Y A345032 Columns k=0..3 give A110654, A000027, A345028, A345029.
%Y A345032 T(n,n) gives A345030.
%Y A345032 Cf. A344821, A345033.
%K A345032 nonn,tabl
%O A345032 1,5
%A A345032 _Seiichi Manyama_, Jun 06 2021