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

%I A191829 #28 Jul 25 2024 10:08:56
%S A191829 0,0,1,6,18,41,78,132,209,306,435,591,780,1008,1268,1584,1917,2335,
%T A191829 2751,3294,3776,4467,5034,5875,6522,7548,8250,9498,10260,11734,12546,
%U A191829 14268,15134,17151,18018,20361,21234,23907,24818,27834,28677,32218,32937,36825,37672,41970,42576,47633,48006,53436,54008,59868,60042,67020,66690
%N A191829 a(n) = Sum_{i+j+k=n, i,j,k >= 1} tau(i)*tau(j)*tau(k), where tau() = A000005().
%C A191829 This is Andrews's D_{0,0,0}(n).
%H A191829 Alois P. Heinz, <a href="/A191829/b191829.txt">Table of n, a(n) for n = 1..10000</a>
%H A191829 George E. Andrews, <a href="http://dx.doi.org/10.1007/BF01608779">Stacked lattice boxes</a>, Ann. Comb. 3 (1999), 115-130.
%H A191829 E. C. Titchmarsh, <a href="https://doi.org/10.1093/qmath/os-13.1.129">Some problems in the analytic theory of numbers</a>, The Quarterly Journal of Mathematics 1 (1942): 129-152.
%F A191829 G.f.: (Sum_{k>=1} x^k/(1 - x^k))^3. - _Ilya Gutkovskiy_, Jan 01 2017
%F A191829 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
%p A191829 with(numtheory);
%p A191829 D000:=proc(n) local t1,i,j;
%p A191829 t1:=0;
%p A191829 for i from 1 to n-1 do
%p A191829 for j from 1 to n-1 do
%p A191829 if (i+j < n) then t1 := t1+numtheory:-tau(i)*numtheory:-tau(j)*numtheory:-tau(n-i-j); fi;
%p A191829 od; od;
%p A191829 t1;
%p A191829 end;
%p A191829 [seq(D000(n),n=1..60)];
%p A191829 # second Maple program:
%p A191829 b:= proc(n, k) option remember; `if`(k=0, `if`(n=0, 1, 0),
%p A191829       `if`(k=1, `if`(n=0, 0, numtheory[tau](n)), (q->
%p A191829        add(b(j, q)*b(n-j, k-q), j=0..n))(iquo(k, 2))))
%p A191829     end:
%p A191829 a:= n-> b(n, 3):
%p A191829 seq(a(n), n=1..55);  # _Alois P. Heinz_, Feb 01 2021
%t A191829 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 *)
%o A191829 (Python)
%o A191829 from sympy import divisor_count
%o A191829 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
%Y A191829 Cf. A000005, A055507, A191831.
%K A191829 nonn,easy
%O A191829 1,4
%A A191829 _N. J. A. Sloane_, Jun 17 2011