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
Keywords
Programs
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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
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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.