|
| |
|
|
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.
|
|
5
|
|
|
|
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
(list;
table;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
|
OFFSET
|
1,3
|
|
|
LINKS
|
|
|
|
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))
|
|
|
CROSSREFS
|
|
|
|
KEYWORD
|
|
|
|
AUTHOR
|
|
|
|
STATUS
|
approved
|
| |
|
|
