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

0,3

Links

Table of n, a(n) for n=0..41.

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.

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]

Crossrefs

Cf. A000009, A000290, A095944, A360486.

Sequence in context: A299903 A299276 A267167 * A023652 A283523 A101648

Adjacent sequences: A360484 A360485 A360486 * A360488 A360489 A360490

Keyword

nonn

Author

Vaclav Kotesovec, Feb 09 2023

Status

approved