A191829 a(n) = Sum_{i+j+k=n, i,j,k >= 1} tau(i)*tau(j)*tau(k), where tau() = A000005().

0, 0, 1, 6, 18, 41, 78, 132, 209, 306, 435, 591, 780, 1008, 1268, 1584, 1917, 2335, 2751, 3294, 3776, 4467, 5034, 5875, 6522, 7548, 8250, 9498, 10260, 11734, 12546, 14268, 15134, 17151, 18018, 20361, 21234, 23907, 24818, 27834, 28677, 32218, 32937, 36825, 37672, 41970, 42576, 47633, 48006, 53436, 54008, 59868, 60042, 67020, 66690

Offset: 1

Author

N. J. A. Sloane, Jun 17 2011

Keywords

• nonn
• easy

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.

Crossrefs

Cf. A000005, A055507, A191831.

Programs

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 *)

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

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