A274352 Convolution of A015723 and A000700.

0, 1, 2, 4, 7, 10, 18, 26, 36, 53, 76, 104, 140, 190, 252, 336, 437, 564, 732, 936, 1186, 1504, 1894, 2366, 2950, 3659, 4520, 5564, 6822, 8330, 10152, 12326, 14906, 17996, 21662, 25996, 31135, 37190, 44314, 52704, 62532, 74036, 87504, 103212, 121496, 142798

Offset: 0

Author

R. J. Mathar, Jun 18 2016

Keywords

• nonn

Comments

Also the convolution of A080054 and A048272.

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..10000
• Mircea Merca, Combinatorial interpretations of a recent convolution for the number of divisors of a positive integer, Journal of Number Theory, Volume 160, March 2016, Pages 60-75, corollary 3.4

Crossrefs

Cf. A015723, A000700, A080054, A048272.

Programs

Maple

with(numtheory):
b:= proc(n) option remember; `if`(n=0, 1, add(add(d*
     [0, 2, -1, 2][1+irem(d, 4)], d=divisors(j))*b(n-j), j=1..n)/n)
    end:
g:= proc(n) option remember; add((-1)^(d+1), d=divisors(n)) end:
a:= n-> add(b(j)*g(n-j), j=0..n):
seq(a(n), n=0..60);  # Alois P. Heinz, Jun 18 2016

Mathematica

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 *)

Formula

a(n) = Sum_{k=0..n} A015723(k)*A000700(n-k).

a(n) ~ log(2) * exp(Pi*sqrt(n/2)) / (Pi * 2^(3/4) * n^(1/4)). - Vaclav Kotesovec, Sep 14 2021