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A319649
Square array A(n,k), n >= 1, k >= 0, read by antidiagonals: A(n,k) = Sum_{j=1..n} j^k * floor(n/j).
6
1, 1, 3, 1, 4, 5, 1, 6, 8, 8, 1, 10, 16, 15, 10, 1, 18, 38, 37, 21, 14, 1, 34, 100, 111, 63, 33, 16, 1, 66, 278, 373, 237, 113, 41, 20, 1, 130, 796, 1335, 999, 489, 163, 56, 23, 1, 258, 2318, 4957, 4461, 2393, 833, 248, 69, 27, 1, 514, 6820, 18831, 20583, 12513, 4795, 1418, 339, 87, 29
OFFSET
1,3
FORMULA
G.f. of column k: (1/(1 - x)) * Sum_{j>=1} j^k*x^j/(1 - x^j).
A(n,k) = Sum_{j=1..n} sigma_k(j).
EXAMPLE
Square array begins:
1, 1, 1, 1, 1, 1, ...
3, 4, 6, 10, 18, 34, ...
5, 8, 16, 38, 100, 278, ...
8, 15, 37, 111, 373, 1335, ...
10, 21, 63, 237, 999, 4461, ...
14, 33, 113, 489, 2393, 12513, ...
MATHEMATICA
Table[Function[k, Sum[j^k Floor[n/j] , {j, 1, n}]][i - n], {i, 0, 11}, {n, 1, i}] // Flatten
Table[Function[k, SeriesCoefficient[1/(1 - x) Sum[j^k x^j/(1 - x^j), {j, 1, n}], {x, 0, n}]][i - n], {i, 0, 11}, {n, 1, i}] // Flatten
Table[Function[k, Sum[DivisorSigma[k, j], {j, 1, n}]][i - n], {i, 0, 11}, {n, 1, i}] // Flatten
PROG
(Python)
from itertools import count, islice
from math import isqrt
from sympy import bernoulli
def A319649_T(n, k): return (((s:=isqrt(n))+1)*(bernoulli(k+1)-bernoulli(k+1, s+1))+sum(w**k*(k+1)*((q:=n//w)+1)-bernoulli(k+1)+bernoulli(k+1, q+1) for w in range(1, s+1)))//(k+1) + int(k==0)
def A319649_gen(): # generator of terms
return (A319649_T(k+1, n-k-1) for n in count(1) for k in range(n))
A319649_list = list(islice(A319649_gen(), 30)) # Chai Wah Wu, Oct 24 2023
CROSSREFS
Columns k=0..5 give A006218, A024916, A064602, A064603, A064604, A248076.
Cf. A082771, A109974, A319194 (diagonal).
Sequence in context: A097207 A266101 A118469 * A198553 A324288 A302917
KEYWORD
nonn,tabl
AUTHOR
Ilya Gutkovskiy, Dec 09 2018
STATUS
approved