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A360486
Convolution of A000041 and A000290.
1
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
0,3
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.
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]
PROG
(PARI) a(n) = sum(k=0, n, numbpart(k)*(n-k)^2); \\ Michel Marcus, Feb 09 2023
CROSSREFS
KEYWORD
nonn
AUTHOR
Vaclav Kotesovec, Feb 09 2023
STATUS
approved