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A344726 Square array T(n,k), n >= 1, k >= 1, read by antidiagonals downwards, where T(n,k) = Sum_{j=1..n} (-1)^(j+1) * floor(n/j)^k.

%I A344726 #26 May 30 2021 06:38:39
%S A344726 1,1,1,1,3,3,1,7,9,2,1,15,27,12,4,1,31,81,56,22,4,1,63,243,240,118,30,
%T A344726 6,1,127,729,992,610,196,44,4,1,255,2187,4032,3094,1230,324,48,7,1,
%U A344726 511,6561,16256,15562,7564,2336,448,71,7,1,1023,19683,65280,77998,45990,16596,3840,685,83,9
%N A344726 Square array T(n,k), n >= 1, k >= 1, read by antidiagonals downwards, where T(n,k) = Sum_{j=1..n} (-1)^(j+1) * floor(n/j)^k.
%H A344726 Seiichi Manyama, <a href="/A344726/b344726.txt">Antidiagonals n = 1..140, flattened</a>
%F A344726 G.f. of column k: (1/(1 - x)) * Sum_{j>=1} (j^k - (j - 1)^k) * x^j/(1 + x^j).
%F A344726 T(n,k) = Sum_{j=1..n} Sum_{d|j} (-1)^(j/d + 1) * (d^k - (d - 1)^k).
%e A344726 Square array begins:
%e A344726   1,  1,   1,    1,    1,     1, ...
%e A344726   1,  3,   7,   15,   31,    63, ...
%e A344726   3,  9,  27,   81,  243,   729, ...
%e A344726   2, 12,  56,  240,  992,  4032, ...
%e A344726   4, 22, 118,  610, 3094, 15562, ...
%e A344726   4, 30, 196, 1230, 7564, 45990, ...
%t A344726 T[n_, k_] := Sum[(-1)^(j + 1) * Quotient[n, j]^k, {j, 1, n}]; Table[T[k, n - k + 1], {n, 1, 11}, {k, 1, n}] // Flatten (* _Amiram Eldar_, May 27 2021 *)
%o A344726 (PARI) T(n, k) = sum(j=1, n, (-1)^(j+1)*(n\j)^k);
%o A344726 (PARI) T(n, k) = sum(j=1, n, sumdiv(j, d, (-1)^(j/d+1)*(d^k-(d-1)^k)));
%Y A344726 Columns k=1..5 give A059851, A344720, A344721, A344722, A344723.
%Y A344726 T(n,n) gives A344724.
%Y A344726 Cf. A344725.
%K A344726 nonn,tabl
%O A344726 1,5
%A A344726 _Seiichi Manyama_, May 27 2021