A327644 Number of proper many times partitions of n.

1, 1, 2, 4, 14, 44, 244, 1196, 9366, 62296, 584016, 5120548, 60244028, 627389924, 8378159376, 106097674780, 1652301306958, 23655318730276, 409987534384504, 6742903763089068, 130675390985884516, 2396246933608687036, 50636625943991790784, 1032841246318579471748

Offset: 0

Author

Alois P. Heinz, Sep 20 2019

Keywords

• nonn

Comments

In each step at least one part is replaced by the partition of itself into smaller parts. The parts are not resorted.

Examples

			a(3) = 4: 3, 3->21, 3->111, 3->21->111.
a(4) = 14: 4, 4->31, 4->22, 4->211, 4->1111, 4->31->211, 4->31->1111, 4->22->112, 4->22->211, 4->22->1111, 4->211->1111, 4->31->211->1111, 4->22->112->1111, 4->22->211->1111.
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..400
• Vaclav Kotesovec, Plot of a(n+1)/(n*a(n)) for n = 1..400

Crossrefs

Row sums of A327639.

Cf. A327648.

Programs

Maple

b:= proc(n, i, k) option remember; `if`(n=0 or k=0, 1, `if`(i>1,
      b(n, i-1, k), 0) +b(i$2, k-1)*b(n-i, min(n-i, i), k))
    end:
a:= n-> add(add(b(n$2, i)*(-1)^(k-i)*
        binomial(k, i), i=0..k), k=0..max(0, n-1)):
seq(a(n), n=0..23);

Mathematica

b[n_, i_, k_] := b[n, i, k] = If[n == 0 || k == 0, 1, If[i > 1, b[n, i - 1, k], 0] + b[i, i, k - 1] b[n - i, Min[n - i, i], k]];
a[n_] := Sum[b[n, n, i] (-1)^(k - i) Binomial[k, i], {k, 0, Max[0, n - 1]}, {i, 0, k}];
a /@ Range[0, 23] (* Jean-François Alcover, Dec 09 2020, after Alois P. Heinz *)