|
%I #20 Sep 14 2021 11:01:38
%S 0,1,2,4,7,10,18,26,36,53,76,104,140,190,252,336,437,564,732,936,1186,
%T 1504,1894,2366,2950,3659,4520,5564,6822,8330,10152,12326,14906,17996,
%U 21662,25996,31135,37190,44314,52704,62532,74036,87504,103212,121496,142798
%N Convolution of A015723 and A000700.
%C Also the convolution of A080054 and A048272.
%H Alois P. Heinz, <a href="/A274352/b274352.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 3.4
%F a(n) = Sum_{k=0..n} A015723(k)*A000700(n-k).
%F a(n) ~ log(2) * exp(Pi*sqrt(n/2)) / (Pi * 2^(3/4) * n^(1/4)). - _Vaclav Kotesovec_, Sep 14 2021
%p with(numtheory):
%p b:= proc(n) option remember; `if`(n=0, 1, add(add(d*
%p [0, 2, -1, 2][1+irem(d, 4)], d=divisors(j))*b(n-j), j=1..n)/n)
%p end:
%p g:= proc(n) option remember; add((-1)^(d+1), d=divisors(n)) end:
%p a:= n-> add(b(j)*g(n-j), j=0..n):
%p seq(a(n), n=0..60); # _Alois P. Heinz_, Jun 18 2016
%t q[n_, k_] := q[n, k] = If[n < k || k < 1, 0, If[n == 1, 1, q[n - k, k] + q[n - k, k - 1]]]; Table[Sum[SeriesCoefficient[Product[1 + x^j, {j, 1, k, 2}], {x, 0, k}] Sum[i q[#, i], {i, 1, Floor[(Sqrt[8 # + 1] - 1)/2]}] &[n - k], {k, 0, n}], {n, 0, 45}] (* _Michael De Vlieger_, Jun 18 2016, after _Vaclav Kotesovec_ at A015723 and _Vladimir Reshetnikov_ at A000700 *)
%Y Cf. A015723, A000700, A080054, A048272.
%K nonn
%O 0,3
%A _R. J. Mathar_, Jun 18 2016
|
