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A319649 Square array A(n,k), n >= 1, k >= 0, read by antidiagonals: A(n,k) = Sum_{j=1..n} j^k * floor(n/j).

%I A319649 #56 Oct 24 2023 15:43:08
%S A319649 1,1,3,1,4,5,1,6,8,8,1,10,16,15,10,1,18,38,37,21,14,1,34,100,111,63,
%T A319649 33,16,1,66,278,373,237,113,41,20,1,130,796,1335,999,489,163,56,23,1,
%U A319649 258,2318,4957,4461,2393,833,248,69,27,1,514,6820,18831,20583,12513,4795,1418,339,87,29
%N A319649 Square array A(n,k), n >= 1, k >= 0, read by antidiagonals: A(n,k) = Sum_{j=1..n} j^k * floor(n/j).
%H A319649 <a href="/index/Si#SIGMAN">Index entries for sequences related to sigma(n)</a>
%F A319649 G.f. of column k: (1/(1 - x)) * Sum_{j>=1} j^k*x^j/(1 - x^j).
%F A319649 A(n,k) = Sum_{j=1..n} sigma_k(j).
%e A319649 Square array begins:
%e A319649    1,   1,    1,    1,     1,      1,  ...
%e A319649    3,   4,    6,   10,    18,     34,  ...
%e A319649    5,   8,   16,   38,   100,    278,  ...
%e A319649    8,  15,   37,  111,   373,   1335,  ...
%e A319649   10,  21,   63,  237,   999,   4461,  ...
%e A319649   14,  33,  113,  489,  2393,  12513,  ...
%t A319649 Table[Function[k, Sum[j^k Floor[n/j] , {j, 1, n}]][i - n], {i, 0, 11}, {n, 1, i}] // Flatten
%t A319649 Table[Function[k, SeriesCoefficient[1/(1 - x) Sum[j^k x^j/(1 - x^j), {j, 1, n}], {x, 0, n}]][i - n], {i, 0, 11}, {n, 1, i}] // Flatten
%t A319649 Table[Function[k, Sum[DivisorSigma[k, j], {j, 1, n}]][i - n], {i, 0, 11}, {n, 1, i}] // Flatten
%o A319649 (Python)
%o A319649 from itertools import count, islice
%o A319649 from math import isqrt
%o A319649 from sympy import bernoulli
%o A319649 def A319649_T(n,k): return (((s:=isqrt(n))+1)*(bernoulli(k+1)-bernoulli(k+1,s+1))+sum(w**k*(k+1)*((q:=n//w)+1)-bernoulli(k+1)+bernoulli(k+1,q+1) for w in range(1,s+1)))//(k+1) + int(k==0)
%o A319649 def A319649_gen(): # generator of terms
%o A319649      return (A319649_T(k+1,n-k-1) for n in count(1) for k in range(n))
%o A319649 A319649_list = list(islice(A319649_gen(),30)) # _Chai Wah Wu_, Oct 24 2023
%Y A319649 Columns k=0..5 give A006218, A024916, A064602, A064603, A064604, A248076.
%Y A319649 Cf. A082771, A109974, A319194 (diagonal).
%K A319649 nonn,tabl
%O A319649 1,3
%A A319649 _Ilya Gutkovskiy_, Dec 09 2018