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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).

%I A327643 #63 Apr 28 2020 09:32:06
%S A327643 1,1,1,3,6,24,84,498,2220,15108,92328,773580,5636460,53563476,
%T A327643 471562512,5270698716,52117937052,637276396764,7317811499736,
%U A327643 100453675122444,1276319138168796,19048874583061716,270233458572751440,4442429353548965628,68384217440167826412
%N 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).
%C A327643 Number of proper (n-1)-times partitions of n, cf. A327639.
%C A327643 Might be called "Half-Factorial numbers" analog to the "Half-Catalan numbers" (A000992).
%C A327643 The recursion formula is a special case of the formula given in A327729.
%C A327643 a(n+1)/(n*a(n)) tends to 0.67617164... - _Vaclav Kotesovec_, Apr 28 2020
%H A327643 Alois P. Heinz, <a href="/A327643/b327643.txt">Table of n, a(n) for n = 1..481</a>
%H A327643 Vaclav Kotesovec, <a href="/A327643/a327643.jpg">Plot of a(n+1)/(n*a(n)) for n = 1..10000</a>
%H A327643 Wikipedia, <a href="https://en.wikipedia.org/wiki/Partition_(number_theory)">Partition (number theory)</a>
%F A327643 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 A327643 a(n) = A327639(n,n-1) = A327631(n,n-1)/n.
%e A327643 a(1) = 1:
%e A327643   1
%e A327643 a(2) = 1:
%e A327643   2 -> 11
%e A327643 a(3) = 1:
%e A327643   3 -> 21 -> 111
%e A327643 a(4) = 3:
%e A327643   4 -> 31 -> 211 -> 1111
%e A327643   4 -> 22 -> 112 -> 1111
%e A327643   4 -> 22 -> 211 -> 1111
%e A327643 a(5) = 6:
%e A327643   5 -> 41 -> 311 -> 2111 -> 11111
%e A327643   5 -> 41 -> 221 -> 1121 -> 11111
%e A327643   5 -> 41 -> 221 -> 2111 -> 11111
%e A327643   5 -> 32 -> 212 -> 1112 -> 11111
%e A327643   5 -> 32 -> 212 -> 2111 -> 11111
%e A327643   5 -> 32 -> 311 -> 2111 -> 11111
%p A327643 b:= proc(n, i, k) option remember; `if`(n=0 or k=0, 1, `if`(i>1,
%p A327643       b(n, i-1, k), 0) +b(i$2, k-1)*b(n-i, min(n-i, i), k))
%p A327643     end:
%p A327643 a:= n-> add(b(n$2, i)*(-1)^(n-1-i)*binomial(n-1, i), i=0..n-1):
%p A327643 seq(a(n), n=1..29);
%p A327643 # second Maple program:
%p A327643 a:= proc(n) option remember; `if`(n=1, 1,
%p A327643       add(a(j)*a(n-j)*binomial(n-2, j-1), j=1..n/2))
%p A327643     end:
%p A327643 seq(a(n), n=1..29);
%t A327643 a[n_] := a[n] = Sum[Binomial[n-2, j-1] a[j] a[n-j], {j, n/2}]; a[1] = 1;
%t A327643 Array[a, 25] (* _Jean-François Alcover_, Apr 28 2020 *)
%Y A327643 Cf. A000142, A000992, A002846 (only one part of each size is replaceable), A327631, A327639, A327697, A327698, A327699, A327702, A327729.
%K A327643 nonn
%O A327643 1,4
%A A327643 _Alois P. Heinz_, Sep 20 2019