A360487 Convolution of A000009 and A000290.

0, 1, 5, 14, 31, 60, 106, 176, 279, 426, 631, 912, 1291, 1795, 2457, 3317, 4424, 5837, 7626, 9875, 12684, 16171, 20476, 25764, 32228, 40094, 49626, 61131, 74966, 91545, 111346, 134921, 162906, 196031, 235134, 281175, 335251, 398615, 472695, 559115, 659721, 776608

Offset: 0

Author

Vaclav Kotesovec, Feb 09 2023

Keywords

• nonn

Crossrefs

Cf. A000009, A000290, A095944, A360486.

Programs

Maple

b:= proc(n) option remember; `if`(n=0, 1, add(b(n-j)*add(
     `if`(d::odd, d, 0), d=numtheory[divisors](j)), j=1..n)/n)
    end:
a:= n-> add(b(n-j)*j^2, j=0..n):
seq(a(n), n=0..42);  # Alois P. Heinz, Feb 09 2023

Mathematica

Table[Sum[PartitionsQ[k]*(n-k)^2, {k, 0, n}], {n, 0, 60}]
CoefficientList[Series[x*(1+x)*QPochhammer[-1, x] / (2*(1-x)^3), {x, 0, 60}], x]

Formula

a(n) = Sum_{k=0..n} A000009(k) * (n-k)^2.

G.f.: x*(1+x)/(1-x)^3 * Product_{k>=1} (1 + x^k).

a(n) ~ 4 * 3^(5/4) * n^(3/4) * exp(sqrt(n/3)*Pi) / Pi^3.