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

0,3

Comments

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

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

Example

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.

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 *)

Crossrefs

Row sums of A327639.

Cf. A327648.

Sequence in context: A007866 A226909 A121751 * A151355 A014272 A070822

Adjacent sequences: A327641 A327642 A327643 * A327645 A327646 A327647

Keyword

nonn

Author

Alois P. Heinz, Sep 20 2019

Status

approved