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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

%I

%S 1,1,1,3,6,24,84,498,2220,15108,92328,773580,5636460,53563476,

%T 471562512,5270698716,52117937052,637276396764,7317811499736,

%U 100453675122444,1276319138168796,19048874583061716,270233458572751440,4442429353548965628,68384217440167826412

%N 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).

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

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

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

%H Alois P. Heinz, <a href="/A327643/b327643.txt">Table of n, a(n) for n = 1..481</a>

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Partition_(number_theory)">Partition (number theory)</a>

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

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

%e a(1) = 1:

%e 1

%e a(2) = 1:

%e 2 -> 11

%e a(3) = 1:

%e 3 -> 21 -> 111

%e a(4) = 3:

%e 4 -> 31 -> 211 -> 1111

%e 4 -> 22 -> 112 -> 1111

%e 4 -> 22 -> 211 -> 1111

%e a(5) = 6:

%e 5 -> 41 -> 311 -> 2111 -> 11111

%e 5 -> 41 -> 221 -> 1121 -> 11111

%e 5 -> 41 -> 221 -> 2111 -> 11111

%e 5 -> 32 -> 212 -> 1112 -> 11111

%e 5 -> 32 -> 212 -> 2111 -> 11111

%e 5 -> 32 -> 311 -> 2111 -> 11111

%p b:= proc(n, i, k) option remember; `if`(n=0 or k=0, 1, `if`(i>1,

%p b(n, i-1, k), 0) +b(i$2, k-1)*b(n-i, min(n-i, i), k))

%p end:

%p a:= n-> add(b(n$2, i)*(-1)^(n-1-i)*binomial(n-1, i), i=0..n-1):

%p seq(a(n), n=1..29);

%p # second Maple program:

%p a:= proc(n) option remember; `if`(n=1, 1,

%p add(a(j)*a(n-j)*binomial(n-2, j-1), j=1..n/2))

%p end:

%p seq(a(n), n=1..29);

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

%K nonn

%O 1,4

%A _Alois P. Heinz_, Sep 20 2019

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Last modified February 27 13:18 EST 2020. Contains 332306 sequences. (Running on oeis4.)