A365821 Total number of partitions of [n-s] whose block minima sum to s, summed over all s.

1, 0, 1, 1, 1, 2, 3, 6, 11, 23, 46, 97, 205, 448, 1004, 2339, 5661, 14291, 37507, 101962, 285386, 817772, 2386946, 7069893, 21195092, 64225525, 196636559, 608551084, 1905848637, 6049696252, 19501015441, 63960251538, 213822965681, 729536174204, 2541833303563

Offset: 0

Author

Alois P. Heinz, Dec 14 2023

Keywords

• nonn

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..350
• Wikipedia, Partition of a set

Crossrefs

Antidiagonal sums of A124327.

Cf. A368102.

Programs

Maple

b:= proc(n, i, t, m) option remember; `if`(n=0, t^(m-i+1),
     `if`((i+m)*(m+1-i)/2n, 0, `if`(t=0, 0,
      t*b(n, i+1, t, m))+ b(n-i, i+1, t+1, m)))
    end:
a:= n-> add(b(k, 1, 0, n-k), k=0..n):
seq(a(n), n=0..42);

Formula

a(7) = 6: 12|3, 134|2, 13|24, 14|23, 1|234, 123456.

a(8) = 11: 124|3, 12|34, 1345|2, 134|25, 135|24, 13|245, 145|23, 14|235, 15|234, 1|2345, 1234567.