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A307756
Exponential convolution of number of partitions into distinct parts (A000009) with themselves.
3
1, 2, 4, 10, 26, 66, 184, 472, 1268, 3340, 8748, 22772, 59102, 151590, 386830, 983914, 2489384, 6263284, 15703204, 39221884, 97498736, 241538472, 596115898, 1465958522, 3595196600, 8788765304, 21421616934, 52080152238, 126268822824, 305365334180, 736770528064
OFFSET
0,2
LINKS
FORMULA
E.g.f.: (Sum_{k>=0} A000009(k)*x^k/k!)^2.
a(n) = Sum_{k=0..n} binomial(n,k)*A000009(k)*A000009(n-k).
a(n) ~ exp(Pi*sqrt(2*n/3)) * 2^(n - 5/2) / (sqrt(3) * n^(3/2)). - Vaclav Kotesovec, May 06 2019
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(binomial(n, j)*b(j)*b(n-j), j=0..n):
seq(a(n), n=0..30); # Alois P. Heinz, Apr 26 2019
MATHEMATICA
nmax = 30; CoefficientList[Series[Sum[PartitionsQ[k] x^k/k!, {k, 0, nmax}]^2, {x, 0, nmax}], x] Range[0, nmax]!
Table[Sum[Binomial[n, k] PartitionsQ[k] PartitionsQ[n - k], {k, 0, n}], {n, 0, 30}]
CROSSREFS
KEYWORD
nonn
AUTHOR
Ilya Gutkovskiy, Apr 26 2019
STATUS
approved