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).

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, 57, 41, 1, 67, 281, 395, 255, 149, 71, 56, 1, 131, 799, 1373, 1025, 555, 205, 103, 69, 1, 259, 2321, 5027, 4503, 2537, 905, 325, 130, 87, 1, 515, 6823, 18965, 20657, 12867, 4945, 1585, 442, 170, 99

Offset

1,3

Links

Table of n, a(n) for n=1..66.

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))
A356124_list = list(islice(A356124_gen(), 30)) # Chai Wah Wu, Oct 24 2023

Crossrefs

Column k=0..4 give A024916, A143127, A143128, A356125, A356126.

T(n,n) gives A356129.

T(n,n+1) gives A356128.

Cf. A279394, A319649, A350106.

Sequence in context: A356045 A242131 A177266 * A272534 A332522 A173386

Adjacent sequences: A356121 A356122 A356123 * A356125 A356126 A356127

Keyword

nonn,tabl

Author

Seiichi Manyama, Jul 27 2022

Status

approved