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

0,4

Links

Alois P. Heinz, Table of n, a(n) for n = 0..1000

R. C. Rhoades, Strongly Unimodal Sequences and Mixed Mock Modular Forms

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

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

Sequence in context: A255879 A171096 A125869 * A114736 A099417 A139463

Adjacent sequences: A059615 A059616 A059617 * A059619 A059620 A059621

Keyword

nice,nonn

Author

Henry Bottomley, Jan 31 2001

Status

approved