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A356124 Square array T(n,k), n >= 1, k >= 0, read by antidiagonals downwards, where T(n,k) = Sum_{j=1..n} j^k * binomial(floor(n/j)+1,2).

%I A356124 #32 Oct 24 2023 12:40:11
%S A356124 1,1,4,1,5,8,1,7,11,15,1,11,19,23,21,1,19,41,47,33,33,1,35,103,125,77,
%T A356124 57,41,1,67,281,395,255,149,71,56,1,131,799,1373,1025,555,205,103,69,
%U A356124 1,259,2321,5027,4503,2537,905,325,130,87,1,515,6823,18965,20657,12867,4945,1585,442,170,99
%N A356124 Square array T(n,k), n >= 1, k >= 0, read by antidiagonals downwards, where T(n,k) = Sum_{j=1..n} j^k * binomial(floor(n/j)+1,2).
%F A356124 G.f. of column k: (1/(1-x)) * Sum_{j>=1} j^k * x^j/(1 - x^j)^2.
%F A356124 T(n,k) = Sum_{j=1..n} j * sigma_{k-1}(j).
%e A356124 Square array begins:
%e A356124    1,  1,   1,   1,    1,     1,     1, ...
%e A356124    4,  5,   7,  11,   19,    35,    67, ...
%e A356124    8, 11,  19,  41,  103,   281,   799, ...
%e A356124   15, 23,  47, 125,  395,  1373,  5027, ...
%e A356124   21, 33,  77, 255, 1025,  4503, 20657, ...
%e A356124   33, 57, 149, 555, 2537, 12867, 68969, ...
%t A356124 T[n_, k_] := Sum[j^k * Binomial[Floor[n/j] + 1, 2], {j, 1, n}]; Table[T[k, n - k], {n, 1, 11}, {k, 1, n}] // Flatten (* _Amiram Eldar_, Jul 28 2022 *)
%o A356124 (PARI) T(n, k) = sum(j=1, n, j^k*binomial(n\j+1, 2));
%o A356124 (PARI) T(n, k) = sum(j=1, n, j*sigma(j, k-1));
%o A356124 (Python)
%o A356124 from itertools import count, islice
%o A356124 from math import isqrt
%o A356124 from sympy import bernoulli
%o A356124 def A356124_T(n,k): return ((s:=isqrt(n))*(s+1)*(bernoulli(k+1)-bernoulli(k+1,s+1))+sum(w**k*(k+1)*((q:=n//w)*(q+1))+(w*(bernoulli(k+1,q+1)-bernoulli(k+1))<<1) for w in range(1,s+1)))//(k+1)>>1
%o A356124 def A356124_gen(): # generator of terms
%o A356124      return (A356124_T(k+1,n-k-1) for n in count(1) for k in range(n))
%o A356124 A356124_list = list(islice(A356124_gen(),30)) # _Chai Wah Wu_, Oct 24 2023
%Y A356124 Column k=0..4 give A024916, A143127, A143128, A356125, A356126.
%Y A356124 T(n,n) gives A356129.
%Y A356124 T(n,n+1) gives A356128.
%Y A356124 Cf. A279394, A319649, A350106.
%K A356124 nonn,tabl
%O A356124 1,3
%A A356124 _Seiichi Manyama_, Jul 27 2022