A274355 Convolution of A048272 and A022567.

0, 1, 2, 5, 9, 15, 27, 42, 65, 99, 148, 214, 308, 435, 605, 839, 1145, 1548, 2080, 2769, 3659, 4812, 6278, 8145, 10518, 13506, 17257, 21961, 27821, 35095, 44117, 55243, 68928, 85735, 106285, 131357, 161893, 198944, 243817, 298060, 363446

Offset: 0

Author

R. J. Mathar, Jun 18 2016

Keywords

• nonn

Comments

Also the convolution of A015723 and A000009.

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 3.5.

Crossrefs

Cf. A048272, A022567, A015723, A000009.

Programs

Maple

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

Mathematica

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 *)

Formula

a(n) = Sum_{k=1..n} A048272(k)*A022567(n-k) = Sum_{k=0..n} A015723(k)*A000009(n-k).

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