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A072706
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Number of unimodal partitions/compositions of n into distinct terms.
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42
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1, 1, 1, 3, 3, 5, 9, 11, 15, 21, 33, 39, 55, 69, 93, 127, 159, 201, 261, 327, 411, 537, 653, 819, 1011, 1257, 1529, 1899, 2331, 2829, 3441, 4179, 5031, 6093, 7305, 8767, 10575, 12573, 14997, 17847, 21223, 25089, 29757, 35055, 41379, 48801, 57285, 67131
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OFFSET
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0,4
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COMMENTS
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Also the number of ways to partition a strict integer partition of n into two unordered blocks. - Gus Wiseman, Dec 31 2019
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LINKS
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FORMULA
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G.f.: 1 + sum(n>=1, 2^(n-1)*q^(n*(n+1)/2) / prod(k=1..n, 1-q^k ) ). [Joerg Arndt, Jan 20 2014]
a(n) ~ c^(1/4) * exp(2*sqrt(c*n)) / (4*sqrt(3*Pi)*n^(3/4)), where c = -polylog(2, -2) = A266576 = 1.436746366883680946362902023893583354... - Vaclav Kotesovec, Sep 22 2019
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EXAMPLE
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a(6)=9 since 6 can be written as 1+2+3, 1+3+2, 1+5, 2+3+1, 2+4, 3+2+1, 4+2, 5+1, or 6, but not for example 1+4+1 (which does not have distinct terms) nor 2+1+3 (which is not unimodal).
The a(10) = 33 such compositions of 10 are:
01: [ 1 2 3 4 ]
02: [ 1 2 4 3 ]
03: [ 1 2 7 ]
04: [ 1 3 4 2 ]
05: [ 1 3 6 ]
06: [ 1 4 3 2 ]
07: [ 1 4 5 ]
08: [ 1 5 4 ]
09: [ 1 6 3 ]
10: [ 1 7 2 ]
11: [ 1 9 ]
12: [ 2 3 4 1 ]
13: [ 2 3 5 ]
14: [ 2 4 3 1 ]
15: [ 2 5 3 ]
16: [ 2 7 1 ]
17: [ 2 8 ]
18: [ 3 4 2 1 ]
19: [ 3 5 2 ]
20: [ 3 6 1 ]
21: [ 3 7 ]
22: [ 4 3 2 1 ]
23: [ 4 5 1 ]
24: [ 4 6 ]
25: [ 5 3 2 ]
26: [ 5 4 1 ]
27: [ 6 3 1 ]
28: [ 6 4 ]
29: [ 7 2 1 ]
30: [ 7 3 ]
31: [ 8 2 ]
32: [ 9 1 ]
33: [ 10 ]
(End)
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MAPLE
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b:= proc(n, i) option remember; `if`(n>i*(i+1)/2, 0, `if`(n=0, 1,
expand(b(n, i-1) +`if`(i>n, 0, x*b(n-i, i-1)))))
end:
a:= n->(p->add(coeff(p, x, i)*ceil(2^(i-1)), i=0..degree(p)))(b(n$2)):
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MATHEMATICA
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b[n_, i_] := b[n, i] = If[n > i*(i + 1)/2, 0, If[n == 0, 1, Expand[b[n, i - 1] + If[i > n, 0, x*b[n - i, i - 1]]]]]; a[n_] := Function[{p}, Sum[Coefficient[p, x, i]*Ceiling[2^(i - 1)], {i, 0, Exponent[p, x]}]][b[n, n]]; Table[a[n], {n, 0, 100}] (* Jean-François Alcover, Jan 16 2015, after Alois P. Heinz *)
Table[If[n==0, 1, Sum[2^(Length[ptn]-1), {ptn, Select[IntegerPartitions[n], UnsameQ@@#&]}]], {n, 0, 15}] (* Gus Wiseman, Dec 31 2019 *)
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PROG
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(PARI) N=66; q='q+O('q^N); Vec( 1 + sum(n=1, N, 2^(n-1)*q^(n*(n+1)/2) / prod(k=1, n, 1-q^k ) ) ) \\ Joerg Arndt, Mar 25 2014
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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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