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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). 9
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,4

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.

LINKS

Alois P. Heinz, Table of n, a(n) for n = 1..481

Wikipedia, Partition (number theory)

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.

EXAMPLE

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

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);

CROSSREFS

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

Sequence in context: A279300 A054718 A132390 * A296215 A152761 A295761

Adjacent sequences:  A327640 A327641 A327642 * A327644 A327645 A327646

KEYWORD

nonn

AUTHOR

Alois P. Heinz, Sep 20 2019

STATUS

approved

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Last modified October 23 06:48 EDT 2019. Contains 328335 sequences. (Running on oeis4.)