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A327643
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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).
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9
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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
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OFFSET
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1,4
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COMMENTS
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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.
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LINKS
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FORMULA
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a(n) = Sum_{j=1..floor(n/2)} C(n-2,j-1) a(j)*a(n-j) for n > 1, a(1) = 1.
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EXAMPLE
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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
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MAPLE
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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);
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MATHEMATICA
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a[n_] := a[n] = Sum[Binomial[n-2, j-1] a[j] a[n-j], {j, n/2}]; a[1] = 1;
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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