A320638 Triangle T(n,k) read by rows: the number of partitions of n into k parts which are divisors of n.

1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 0, 0, 0, 1, 1, 1, 2, 2, 1, 1, 1, 0, 0, 0, 0, 0, 1, 1, 1, 1, 2, 2, 1, 1, 1, 1, 0, 1, 0, 1, 0, 1, 0, 1, 1, 1, 0, 1, 2, 2, 1, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 3, 7, 8, 8, 6, 4, 3, 2, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1

Offset

1,18

Links

Alois P. Heinz, Rows n = 1..200, flattened

Formula

T(n,n) = T(n,1) = 1, representing partitions into the trivial divisors.

Example

The triangle starts
1
1 1
1 0 1
1 1 1 1
1 0 0 0 1
1 1 2 2 1 1
1 0 0 0 0 0 1
1 1 1 2 2 1 1 1
1 0 1 0 1 0 1 0 1
1 1 0 1 2 2 1 1 1 1
1 0 0 0 0 0 0 0 0 0 1
1 1 3 7 8 8 6 4 3 2 1 1
1 0 0 0 0 0 0 0 0 0 0 0 1
1 1 0 0 1 1 2 2 1 1 1 1 1 1
1 0 1 0 3 0 3 0 2 0 2 0 1 0 1
1 1 1 2 3 4 4 4 4 3 2 2 2 1 1 1
1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1
1 1 2 3 6 8 9 10 9 8 6 5 4 3 2 2 1 1
1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1
1 1 1 3 6 8 10 10 9 9 8 6 5 4 3 3 2 1 1 1
1 0 1 0 1 0 3 0 3 0 2 0 2 0 2 0 1 0 1 0 1
1 1 0 0 0 0 1 1 1 1 2 2 1 1 1 1 1 1 1 1 1 1
1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1
1 1 3 9 20 33 44 50 51 48 42 36 29 23 18 14 11 8 6 4 3 2 1 1

Maple

A320638 := proc(n, noprts)
    local a, p, w, el ;
    a := 0 ;
    for p in combinat[partition](n) do
        if nops(p) = noprts then
            w := true ;
            for el in p do
                if modp(n, el) <> 0 then
                    w := false;
                    break;
                end if;
            end do:
            if w then
                a := a+1 ;
            end if;
        end if ;
    end do:
    a ;
end proc:
seq(seq(A320638(n, k), k=1..n), n=1..13) ;
# second Maple program:
T:= proc(n) option remember; local b, l;
      l, b:= numtheory[divisors](n),
      proc(m, i) option remember; expand(`if`(m=0, 1,
        `if`(i<1, 0, b(m, i-1)+`if`(l[i]>m, 0, x*b(m-l[i], i)))))
      end; (p-> seq(coeff(p, x, i), i=1..n))(b(n, nops(l)))
    end:
seq(T(n), n=1..20);  # Alois P. Heinz, Nov 08 2023

Mathematica

T[n_, k_] := IntegerPartitions[n, {k}, Divisors[n]] // Length;
Table[T[n, k], {n, 1, 13}, {k, 1, n}] // Flatten (* Jean-François Alcover, Nov 08 2023 *)

Crossrefs

Cf. A018818 (row sums), A008284.

T(2n,n) gives A367189.

Sequence in context: A004562 A261817 A123550 * A262045 A263087 A204433

Adjacent sequences: A320635 A320636 A320637 * A320639 A320640 A320641

Keyword

nonn,look,tabl

Author

R. J. Mathar, Oct 18 2018

Status

approved