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

0,3

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

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

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 *)

Crossrefs

Cf. A015723, A000700, A080054, A048272.

Sequence in context: A032705 A332639 A281168 * A343328 A088762 A217385

Adjacent sequences: A274349 A274350 A274351 * A274353 A274354 A274355

Keyword

nonn

Author

R. J. Mathar, Jun 18 2016

Status

approved