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%I #12 Oct 09 2018 07:28:17
%S 0,1,2,5,9,15,27,42,65,99,148,214,308,435,605,839,1145,1548,2080,2769,
%T 3659,4812,6278,8145,10518,13506,17257,21961,27821,35095,44117,55243,
%U 68928,85735,106285,131357,161893,198944,243817,298060,363446
%N Convolution of A048272 and A022567.
%C Also the convolution of A015723 and A000009.
%H Alois P. Heinz, <a href="/A274355/b274355.txt">Table of n, a(n) for n = 0..10000</a>
%H Mircea Merca, <a href="http://dx.doi.org/10.1016/j.jnt.2015.08.014">Combinatorial interpretations of a recent convolution for the number of divisors of a positive integer</a>, Journal of Number Theory, Volume 160, March 2016, Pages 60-75, corollary 3.5.
%F a(n) = Sum_{k=1..n} A048272(k)*A022567(n-k) = Sum_{k=0..n} A015723(k)*A000009(n-k).
%F a(n) ~ 3^(1/4) * log(2) * exp(Pi*sqrt(2*n/3)) / (2^(7/4) * Pi * n^(1/4)). - _Vaclav Kotesovec_, Oct 09 2018
%p b:= proc(n) option remember; `if`(n=0, 1, add(add(d*[0, 1][1+
%p irem(d, 2)], d=numtheory[divisors](j))*b(n-j), j=1..n)/n)
%p end:
%p g:= proc(n, i) option remember; `if`(i*(i+1)/2<n, 0, `if`(n=0, [1, 0],
%p add((l->[l[1], l[2]+l[1]*j])(g(n-i*j, i-1)), j=0..min(n/i, 1))))
%p end:
%p a:= n-> add(b(n-j)*g(j$2)[2], j=0..n):
%p seq(a(n), n=0..60); # _Alois P. Heinz_, Jun 18 2016
%t Table[Sum[Count[#, _?OddQ] - Count[#, _?EvenQ] &@ Divisors@ k SeriesCoefficient[QPochhammer[q, q^2]^-2, {q, 0, #}] &[n - k], {k, n}], {n, 0, 40}] (* _Michael De Vlieger_, Jun 18 2016, after _Michael Somos_ at A022567 *)
%Y Cf. A048272, A022567, A015723, A000009.
%K nonn
%O 0,3
%A _R. J. Mathar_, Jun 18 2016
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