A304991 -id:A304991 - cp's OEIS frontend

A265096 a(n) = Sum_{k=0..n} p(k)*q(k), where p(k) = partition numbers (A000041) and q(k) = partition numbers into distinct parts (A000009).

1, 2, 4, 10, 20, 41, 85, 160, 292, 532, 952, 1624, 2779, 4597, 7567, 12319, 19711, 30997, 48707, 75167, 115295, 175487, 264665, 395185, 587335, 865371, 1267311, 1845231, 2670627, 3839267, 5498051, 7824331, 11080441, 15624505, 21927225, 30633780, 42642416

Offset: 0

Vaclav Kotesovec, Dec 01 2015

nonn

Cf. A000009, A000041, A000070, A015128, A036469.

Partial sums of A304991.

Table[Sum[PartitionsQ[k]*PartitionsP[k], {k,0,n}], {n,0,50}]

a(n) ~ (sqrt(2)-1) * exp((1+sqrt(2))*Pi*sqrt(n/3)) / (8*3^(1/4)*Pi*n^(5/4)).

A363252 a(n) = gcd(A000041(n), A000009(n)).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 5, 2, 2, 2, 4, 1, 1, 1, 1, 1, 1, 1, 2, 1, 4, 1, 1, 1, 2, 3, 2, 2, 1, 4, 2, 3, 7, 2, 3, 1, 1, 1, 1, 21, 21, 2, 1, 1, 2, 6, 14, 3, 1, 2, 1, 1, 1, 1, 2, 1, 3, 4, 4, 17, 1, 2, 1, 2, 2, 4, 1, 3, 5, 1, 1, 1, 1, 2, 2, 1, 1, 2, 2, 4, 1, 1, 1, 2, 11, 2

Offset: 0

Vaclav Kotesovec, May 23 2023, inspired by Zhi-Wei Sun

nonn

Cf. A000009, A000041, A051177, A056848, A093952, A304991.

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-> igcd(b(n), combinat[numbpart](n)):
seq(a(n), n=0..120);  # Alois P. Heinz, May 23 2023

Table[GCD[PartitionsP[n], PartitionsQ[n]], {n, 0, 100}]

cp's OEIS Frontend

A265096 a(n) = Sum_{k=0..n} p(k)*q(k), where p(k) = partition numbers (A000041) and q(k) = partition numbers into distinct parts (A000009).

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