OFFSET
1,2
LINKS
Seiichi Manyama, Rows n = 1..140, flattened
FORMULA
Triangle read by rows, partial sums of A051731 starting from the right. A051731 as a lower triangular matrix times an all 1's lower triangular matrix.
From Seiichi Manyama, Jan 07 2023: (Start)
G.f. of column k: Sum_{j>=1} x^(k*j)/(1 - x^j).
G.f. of column k: Sum_{j>=k} x^j/(1 - x^j). (End)
Sum_{j=1..n} T(j, k) ~ n * (log(n) + 2*gamma - 1 - H(k-1)), where gamma is Euler's constant (A001620), and H(k) = A001008(k)/A002805(k) is the k-th harmonic number. - Amiram Eldar, Jan 08 2024
EXAMPLE
First few rows of the triangle:
1;
2, 1;
2, 1, 1;
3, 2, 1, 1;
2, 1, 1, 1, 1;
4, 3, 2, 1, 1, 1;
2, 1, 1, 1, 1, 1, 1;
4, 3, 2, 2, 1, 1, 1, 1;
3, 2, 2, 1, 1, 1, 1, 1, 1;
4, 3, 2, 2, 2, 1, 1, 1, 1, 1;
2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1;
6, 5, 4, 3, 2, 2, 1, 1, 1, 1, 1, 1;
...
MAPLE
with(numtheory);
f1:=proc(n) local d, s1, t1, t2, i;
d:=tau(n);
s1:=sort(divisors(n));
t1:=Array(1..n, 0);
for i from 1 to d do t1[n-s1[i]+1]:=1; od:
t2:=PSUM(convert(t1, list));
[seq(t2[n+1-i], i=1..n)];
end proc;
for n from 1 to 15 do lprint(f1(n)); od: # N. J. A. Sloane, Nov 09 2018
MATHEMATICA
T[n_, k_] := DivisorSum[n, Boole[# >= k]&];
Table[T[n, k], {n, 1, 15}, {k, 1, n}] // Flatten (* Jean-François Alcover, Feb 15 2023 *)
PROG
(PARI) row(n) = my(d=divisors(n)); vector(n, k, #select(x->(x>=k), d)); \\ Michel Marcus, Jul 23 2022
CROSSREFS
KEYWORD
AUTHOR
Gary W. Adamson, Oct 30 2007
EXTENSIONS
Clearer definition from N. J. A. Sloane, Nov 09 2018
STATUS
approved