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.

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, 52, 16, 1, 129, 731, 1058, 644, 254, 66, 20, 1, 257, 2189, 4162, 3160, 1396, 382, 92, 23, 1, 513, 6563, 16514, 15692, 8054, 2502, 596, 115, 27, 1, 1025, 19685, 65794, 78256, 47452, 17086, 4388, 833, 147, 29

Offset

1,3

Links

Seiichi Manyama, Antidiagonals n = 1..140, flattened

Formula

G.f. of column k: (1/(1 - x)) * Sum_{j>=1} (j^k - (j - 1)^k) * x^j/(1 - x^j).

T(n,k) = Sum_{j=1..n} Sum_{d|j} d^k - (d - 1)^k.

Example

Square array begins:
   1,  1,   1,    1,    1,     1, ...
   3,  5,   9,   17,   33,    65, ...
   5, 11,  29,   83,  245,   731, ...
   8, 22,  74,  274, 1058,  4162, ...
  10, 32, 136,  644, 3160, 15692, ...
  14, 52, 254, 1396, 8054, 47452, ...

Mathematica

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 *)

Prog

(PARI) T(n, k) = sum(j=1, n, (n\j)^k);
(PARI) T(n, k) = sum(j=1, n, sumdiv(j, d, d^k-(d-1)^k));
(Python)
from math import isqrt
from itertools import count, islice
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))
def A344725_gen(): # generator of terms
     return (A344725_T(k+1, n-k) for n in count(1) for k in range(n))
A344725_list = list(islice(A344725_gen(), 30)) # Chai Wah Wu, Oct 26 2023

Crossrefs

Columns k=1..5 give A006218, A222548, A318742, A318743, A318744.

T(n,n) gives A332469.

Cf. A306533, A344726.

Sequence in context: A209159 A182397 A343510 * A209560 A211977 A072919

Adjacent sequences: A344722 A344723 A344724 * A344726 A344727 A344728

Keyword

nonn,tabl

Author

Seiichi Manyama, May 27 2021

Status

approved