A370368 Total sum over all j in [n] of the number of partitions of [j*(n-j)] into (n-j) sets of size j having no set of consecutive numbers whose maximum (if j>0) is a multiple of j.

1, 1, 1, 1, 3, 18, 347, 20679, 4064088, 3206794270, 9817417580226, 147957639234186793, 9515125170594095021483, 3369265619091187775505912588, 5792039079391869138256364232105952, 55416702792637442337898498177490975722265

Offset: 0

Author

Alois P. Heinz, Feb 16 2024

Keywords

• nonn

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..54
• Wikipedia, Partition of a set

Crossrefs

Antidiagonal sums of A370366.

Cf. A370365, A370407.

Programs

Maple

b:= proc(n, k) `if`(k=0, `if`(n=0, 1, 0), add(
      (-1)^(n-j)*binomial(n, j)*(k*j)!/(j!*k!^j), j=0..n))
    end:
a:= n-> add(b(j, n-j), j=0..n):
seq(a(n), n=0..15);

Formula

a(n) = Sum_{j=0..n} A370366(j,n-j).

a(n) = A370407(n) - A370365(n).