A390030 - OEIS

A390030

Number of perfect overpartitions of n.

1, 2, 1, 5, 1, 5, 1, 13, 2, 5, 1, 19, 1, 5, 3, 34, 1, 13, 1, 19, 3, 5, 1, 65, 2, 5, 4, 19, 1, 21, 1, 89, 3, 5, 3, 61, 1, 5, 3, 65, 1, 21, 1, 19, 8, 5, 1, 210, 2, 13, 3, 19, 1, 32, 3, 65, 3, 5, 1, 103, 1, 5, 8, 233, 3, 21, 1, 19, 3, 21, 1, 248, 1, 5, 8, 19, 3, 21, 1, 210, 8, 5, 1, 103, 3, 5, 3, 65, 1, 70, 3

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OFFSET

0,2

COMMENTS

A perfect overpartition of n is one that contains exactly one overpartition of every positive integer less than n.

LINKS

Table of n, a(n) for n=0..90.

FORMULA

a(n) = Sum_{r=0..s} Sum_{v=r..s} binomial(v,r)*f_v(n+1), where s = A007814(n+1) and f_v(N) is the number of ordered factorizations of N having v copies of 2; that is, f_v(N) is the number of objects counted by A074206(N) that contain v copies of 2.

EXAMPLE

n=0: 1 (the empty partition)

n=1: 2 (1), (1')

n=2: 1 (1,1)

n=3: 5 (1,1,1), (2,1), (2',1), (2,1'), (2',1')

n=4: 1 (1,1,1,1)

n=5: 5 (1,1,1,1,1), (2,2,1), (2,2,1'), (3,1,1), (3',1,1)

n=6: 1 (1,1,1,1,1,1)

n=7: 13 (1,1,1,1,1,1,1), (2,2,2,1), (2,2,2,1'), (4,1,1,1), (4',1,1,1), (2,4,1), (2',4,1), (2,4',1), (2,4,1'), (2',4',1), (2',4,1'), (2,4',1'), (2',4',1')

CROSSREFS

Cf. A015128, A002033, A330773, A074206.

Sequence in context: A369741 A014652 A385597 * A379654 A060448 A378543

Adjacent sequences: A390027 A390028 A390029 * A390031 A390032 A390033

KEYWORD

nonn

AUTHOR

Augustine O. Munagi, Oct 23 2025

STATUS

approved