OFFSET
1,4
COMMENTS
This is Andrews's D_{0,0,0}(n).
LINKS
Alois P. Heinz, Table of n, a(n) for n = 1..10000
George E. Andrews, Stacked lattice boxes, Ann. Comb. 3 (1999), 115-130.
E. C. Titchmarsh, Some problems in the analytic theory of numbers, The Quarterly Journal of Mathematics 1 (1942): 129-152.
FORMULA
G.f.: (Sum_{k>=1} x^k/(1 - x^k))^3. - Ilya Gutkovskiy, Jan 01 2017
a(n) = Sum_{k=1..n-1} Sum_{i=1..k-1} tau(i)*tau(n-k)*tau(k-i). - Ridouane Oudra, Oct 30 2023
MAPLE
with(numtheory);
D000:=proc(n) local t1, i, j;
t1:=0;
for i from 1 to n-1 do
for j from 1 to n-1 do
if (i+j < n) then t1 := t1+numtheory:-tau(i)*numtheory:-tau(j)*numtheory:-tau(n-i-j); fi;
od; od;
t1;
end;
[seq(D000(n), n=1..60)];
# second Maple program:
b:= proc(n, k) option remember; `if`(k=0, `if`(n=0, 1, 0),
`if`(k=1, `if`(n=0, 0, numtheory[tau](n)), (q->
add(b(j, q)*b(n-j, k-q), j=0..n))(iquo(k, 2))))
end:
a:= n-> b(n, 3):
seq(a(n), n=1..55); # Alois P. Heinz, Feb 01 2021
MATHEMATICA
nmax = 50; Rest[CoefficientList[Series[(-1/2 + (Log[1-x] + QPolyGamma[0, 1, 1/x])/Log[x])^3, {x, 0, nmax}], x]] (* Vaclav Kotesovec, Jan 01 2017 *)
PROG
(Python)
from sympy import divisor_count
def A191829(n): return sum(divisor_count(i)*sum(divisor_count(j)*divisor_count(n-i-j) for j in range(1, n-i)) for i in range(1, n-1)) # Chai Wah Wu, Jul 25 2024
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
N. J. A. Sloane, Jun 17 2011
STATUS
approved