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A344725 Square array T(n,k), n >= 1, k >= 1, read by antidiagonals downwards, where T(n,k) = Sum_{j=1..n} floor(n/j)^k.

%I A344725 #39 Oct 27 2023 11:06:46
%S A344725 1,1,3,1,5,5,1,9,11,8,1,17,29,22,10,1,33,83,74,32,14,1,65,245,274,136,
%T A344725 52,16,1,129,731,1058,644,254,66,20,1,257,2189,4162,3160,1396,382,92,
%U A344725 23,1,513,6563,16514,15692,8054,2502,596,115,27,1,1025,19685,65794,78256,47452,17086,4388,833,147,29
%N A344725 Square array T(n,k), n >= 1, k >= 1, read by antidiagonals downwards, where T(n,k) = Sum_{j=1..n} floor(n/j)^k.
%H A344725 Seiichi Manyama, <a href="/A344725/b344725.txt">Antidiagonals n = 1..140, flattened</a>
%F A344725 G.f. of column k: (1/(1 - x)) * Sum_{j>=1} (j^k - (j - 1)^k) * x^j/(1 - x^j).
%F A344725 T(n,k) = Sum_{j=1..n} Sum_{d|j} d^k - (d - 1)^k.
%e A344725 Square array begins:
%e A344725    1,  1,   1,    1,    1,     1, ...
%e A344725    3,  5,   9,   17,   33,    65, ...
%e A344725    5, 11,  29,   83,  245,   731, ...
%e A344725    8, 22,  74,  274, 1058,  4162, ...
%e A344725   10, 32, 136,  644, 3160, 15692, ...
%e A344725   14, 52, 254, 1396, 8054, 47452, ...
%t A344725 T[n_, k_] := Sum[Quotient[n, j]^k, {j, 1, n}]; Table[T[k, n - k + 1], {n, 1, 10}, {k, 1, n}] // Flatten (* _Amiram Eldar_, May 27 2021 *)
%o A344725 (PARI) T(n, k) = sum(j=1, n, (n\j)^k);
%o A344725 (PARI) T(n, k) = sum(j=1, n, sumdiv(j, d, d^k-(d-1)^k));
%o A344725 (Python)
%o A344725 from math import isqrt
%o A344725 from itertools import count, islice
%o A344725 def A344725_T(n,k): return -(s:=isqrt(n))**(k+1)+sum((q:=n//w)*(w**k-(w-1)**k+q**(k-1)) for w in range(1,s+1))
%o A344725 def A344725_gen(): # generator of terms
%o A344725      return (A344725_T(k+1,n-k) for n in count(1) for k in range(n))
%o A344725 A344725_list = list(islice(A344725_gen(),30)) # _Chai Wah Wu_, Oct 26 2023
%Y A344725 Columns k=1..5 give A006218, A222548, A318742, A318743, A318744.
%Y A344725 T(n,n) gives A332469.
%Y A344725 Cf. A306533, A344726.
%K A344725 nonn,tabl
%O A344725 1,3
%A A344725 _Seiichi Manyama_, May 27 2021