A327643 Number of refinement sequences n -> ... -> {1}^n, where in each step one part is replaced by a partition of itself into two smaller parts (in weakly decreasing order).

1, 1, 1, 3, 6, 24, 84, 498, 2220, 15108, 92328, 773580, 5636460, 53563476, 471562512, 5270698716, 52117937052, 637276396764, 7317811499736, 100453675122444, 1276319138168796, 19048874583061716, 270233458572751440, 4442429353548965628, 68384217440167826412

Offset: 1

Author

Alois P. Heinz, Sep 20 2019

Keywords

• nonn

Comments

Number of proper (n-1)-times partitions of n, cf. A327639.

Might be called "Half-Factorial numbers" analog to the "Half-Catalan numbers" (A000992).

The recursion formula is a special case of the formula given in A327729.

a(n+1)/(n*a(n)) tends to 0.67617164... - Vaclav Kotesovec, Apr 28 2020

Examples

			a(1) = 1:
  1
a(2) = 1:
  2 -> 11
a(3) = 1:
  3 -> 21 -> 111
a(4) = 3:
  4 -> 31 -> 211 -> 1111
  4 -> 22 -> 112 -> 1111
  4 -> 22 -> 211 -> 1111
a(5) = 6:
  5 -> 41 -> 311 -> 2111 -> 11111
  5 -> 41 -> 221 -> 1121 -> 11111
  5 -> 41 -> 221 -> 2111 -> 11111
  5 -> 32 -> 212 -> 1112 -> 11111
  5 -> 32 -> 212 -> 2111 -> 11111
  5 -> 32 -> 311 -> 2111 -> 11111
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 1..481
• Vaclav Kotesovec, Plot of a(n+1)/(n*a(n)) for n = 1..10000
• Wikipedia, Partition (number theory)

Crossrefs

Cf. A000142, A000992, A002846 (only one part of each size is replaceable), A327631, A327639, A327697, A327698, A327699, A327702, A327729.

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(b(n$2, i)*(-1)^(n-1-i)*binomial(n-1, i), i=0..n-1):
seq(a(n), n=1..29);
# second Maple program:
a:= proc(n) option remember; `if`(n=1, 1,
      add(a(j)*a(n-j)*binomial(n-2, j-1), j=1..n/2))
    end:
seq(a(n), n=1..29);

Mathematica

a[n_] := a[n] = Sum[Binomial[n-2, j-1] a[j] a[n-j], {j, n/2}]; a[1] = 1;
Array[a, 25] (* Jean-François Alcover, Apr 28 2020 *)

Formula

a(n) = Sum_{j=1..floor(n/2)} C(n-2,j-1) a(j)*a(n-j) for n > 1, a(1) = 1.

a(n) = A327639(n,n-1) = A327631(n,n-1)/n.