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A059618
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Number of strongly unimodal partitions of n (strongly unimodal means strictly increasing then strictly decreasing).
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12
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1, 1, 1, 3, 4, 6, 10, 15, 21, 30, 43, 59, 82, 111, 148, 199, 263, 344, 451, 584, 751, 965, 1230, 1560, 1973, 2483, 3110, 3885, 4834, 5990, 7405, 9123, 11202, 13724, 16762, 20417, 24815, 30081, 36377, 43900, 52860, 63511, 76166, 91157, 108886, 129842
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OFFSET
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0,4
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LINKS
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FORMULA
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G.f.: sum(k>=0, x^k * prod(i=1..k-1, 1 + x^i)^2 ). - Vladeta Jovovic, Dec 05 2003
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EXAMPLE
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a(6) = 10 since 6 can be written as 6, 5+1, 4+2, 3+2+1, 2+4, 2+3+1, 1+5, 1+4+1, 1+3+2 or 1+2+3 (but for example neither 2+2+1+1 nor 1+2+2+1 which are only weakly unimodal).
The a(7) = 15 strongly unimodal compositions of 7 are
[ #] composition
[ 1] [ 1 2 3 1 ]
[ 2] [ 1 2 4 ]
[ 3] [ 1 3 2 1 ]
[ 4] [ 1 4 2 ]
[ 5] [ 1 5 1 ]
[ 6] [ 1 6 ]
[ 7] [ 2 3 2 ]
[ 8] [ 2 4 1 ]
[ 9] [ 2 5 ]
[10] [ 3 4 ]
[11] [ 4 2 1 ]
[12] [ 4 3 ]
[13] [ 5 2 ]
[14] [ 6 1 ]
[15] [ 7 ]
(End)
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MAPLE
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b:= proc(n, i, t) option remember; `if`(t=0 and n>i*(i-1)/2, 0,
`if`(n=0, 1, add(b(n-j, j, 0), j=1..min(n, i-1))+
`if`(t=1, add(b(n-j, j, 1), j=i+1..n), 0)))
end:
a:= n-> b(n, 0, 1):
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MATHEMATICA
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s[n_?Positive, k_] := s[n, k] = Sum[s[n - k, j], {j, 0, k - 1}]; s[0, 0] = 1; s[0, _] = 0; s[_?Negative, _] = 0; t[n_, k_] := t[n, k] = s[n, k] + Sum[t[n - k, j], {j, k + 1, n}]; a[n_] := t[n, 0]; Table[a[n], {n, 0, 45}] (* Jean-François Alcover, Dec 06 2012, after Vladeta Jovovic *)
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PROG
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(PARI) N=66; x='x+O('x^N); Vec(sum(n=0, N, x^(n) * prod(k=1, n-1, 1+x^k)^2)) \\ Joerg Arndt, Mar 26 2014
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CROSSREFS
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KEYWORD
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nice,nonn
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AUTHOR
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STATUS
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approved
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