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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. a(n+1)/(n*a(n)) tends to 0.67617164... - Vaclav Kotesovec, Apr 28 2020 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) 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); 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 *) CROSSREFS Cf. A000142, A000992, A002846 (only one part of each size is replaceable), A327631, A327639, A327697, A327698, A327699, A327702, A327729. Sequence in context: A054718 A132390 A363016 * 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 November 30 21:14 EST 2023. Contains 367462 sequences. (Running on oeis4.)