A360486 Convolution of A000041 and A000290.

0, 1, 5, 15, 36, 76, 147, 267, 462, 769, 1240, 1947, 2988, 4496, 6649, 9683, 13909, 19734, 27686, 38447, 52892, 72138, 97604, 131084, 174835, 231687, 305173, 399687, 520675, 674865, 870540, 1117869, 1429298, 1820018, 2308521, 2917260, 3673428, 4609885, 5766245

Offset: 0

Author

Vaclav Kotesovec, Feb 09 2023

Keywords

• nonn

Crossrefs

Cf. A000041, A000290, A014153, A360487.

Programs

Maple

a:= n-> add(combinat[numbpart](n-j)*j^2, j=0..n):
seq(a(n), n=0..42);  # Alois P. Heinz, Feb 09 2023

Mathematica

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

PARI

a(n) = sum(k=0, n, numbpart(k)*(n-k)^2); \\ Michel Marcus, Feb 09 2023

Formula

a(n) = Sum_{k=0..n} A000041(k) * (n-k)^2.

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

a(n) ~ 3 * sqrt(2*n) * exp(sqrt(2*n/3)*Pi) / Pi^3.