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URL: https://oeis.org/A059618

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A059618
Number of strongly unimodal partitions of n (strongly unimodal means strictly increasing then strictly decreasing).
12
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
OFFSET
0,4
FORMULA
a(n) = A059619(n,0) = Sum_k A059619(n,k) for k>0 when n>0.
G.f.: sum(k>=0, x^k * prod(i=1..k-1, 1 + x^i)^2 ). - Vladeta Jovovic, Dec 05 2003
EXAMPLE
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).
From Joerg Arndt, Dec 09 2012: (Start)
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)
MAPLE
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):
seq(a(n), n=0..60); # Alois P. Heinz, Mar 21 2014
MATHEMATICA
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 *)
PROG
(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
CROSSREFS
KEYWORD
nice,nonn
AUTHOR
Henry Bottomley, Jan 31 2001
STATUS
approved