OFFSET
1,3
FORMULA
G.f. of column k: (1/(1-x)) * Sum_{j>=1} j^k * x^j/(1 - x^j)^2.
T(n,k) = Sum_{j=1..n} j * sigma_{k-1}(j).
EXAMPLE
Square array begins:
1, 1, 1, 1, 1, 1, 1, ...
4, 5, 7, 11, 19, 35, 67, ...
8, 11, 19, 41, 103, 281, 799, ...
15, 23, 47, 125, 395, 1373, 5027, ...
21, 33, 77, 255, 1025, 4503, 20657, ...
33, 57, 149, 555, 2537, 12867, 68969, ...
MATHEMATICA
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 *)
PROG
(PARI) T(n, k) = sum(j=1, n, j^k*binomial(n\j+1, 2));
(PARI) T(n, k) = sum(j=1, n, j*sigma(j, k-1));
(Python)
from itertools import count, islice
from math import isqrt
from sympy import bernoulli
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
def A356124_gen(): # generator of terms
return (A356124_T(k+1, n-k-1) for n in count(1) for k in range(n))
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Seiichi Manyama, Jul 27 2022
STATUS
approved