A307754 Exponential convolution of number of partitions (A000041) and number of partitions into distinct parts (A000009).

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

Vaclav Kotesovec, Table of n, a(n) for n = 0..3000

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

Cf. A000009, A000041, A015128.

Sequence in context: A105641 A027035 A102406 * A151409 A339289 A003054

Adjacent sequences: A307751 A307752 A307753 * A307755 A307756 A307757

Keyword

nonn

Author

Ilya Gutkovskiy, Apr 26 2019

Status

approved