A059618 Number of strongly unimodal partitions of n (strongly unimodal means strictly increasing then strictly decreasing).

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

Author

Henry Bottomley, Jan 31 2001

Keywords

• nice
• nonn

Examples

			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)
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..1000
• R. C. Rhoades, Strongly Unimodal Sequences and Mixed Mock Modular Forms

Crossrefs

Cf. A000009, A000041, A001523, A059607, A059619.

Programs

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 *)

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

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