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A307754
Exponential convolution of number of partitions (A000041) and number of partitions into distinct parts (A000009).
1
1, 2, 5, 14, 39, 115, 330, 958, 2708, 7700, 21474, 59732, 164126, 447995, 1211720, 3258896, 8699731, 23100190, 60985784, 160172634, 418505039, 1088396720, 2817751182, 7262961898, 18647741599, 47690238685, 121529569106, 308617383361, 781131731981, 1970766138399, 4957228945138
OFFSET
0,2
LINKS
FORMULA
E.g.f.: (Sum_{i>=0} A000041(i)*x^i/i!) * (Sum_{j>=0} A000009(j)*x^j/j!).
a(n) = Sum_{k=0..n} binomial(n,k)*A000041(k)*A000009(n-k).
a(n) ~ exp(Pi*(sqrt(2) + 1)*sqrt(n/6) + ((sqrt(2) - 1)*Pi)^2/48) * 2^(n - 9/4) / (3^(3/4) * n^(7/4)). - Vaclav Kotesovec, May 07 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)*combinat[numbpart](n-j), j=0..n):
seq(a(n), n=0..30); # Alois P. Heinz, Apr 26 2019
MATHEMATICA
nmax = 30; CoefficientList[Series[Sum[PartitionsP[i] x^i/i!, {i, 0, nmax}] Sum[PartitionsQ[j] x^j/j!, {j, 0, nmax}], {x, 0, nmax}], x] Range[0, nmax]!
Table[Sum[Binomial[n, k] PartitionsP[k] PartitionsQ[n - k], {k, 0, n}], {n, 0, 30}]
CROSSREFS
KEYWORD
nonn
AUTHOR
Ilya Gutkovskiy, Apr 26 2019
STATUS
approved