A059619 -id:A059619 - cp's OEIS frontend

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

Henry Bottomley, Jan 31 2001

			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)

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

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

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

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 *)

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

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

A059623 As upper right triangle, number of weakly unimodal partitions of n where initial part is k (n >= k >= 1).

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 4, 2, 1, 1, 8, 3, 2, 1, 1, 15, 5, 3, 2, 1, 1, 27, 8, 5, 3, 2, 1, 1, 47, 13, 7, 5, 3, 2, 1, 1, 79, 21, 11, 7, 5, 3, 2, 1, 1, 130, 33, 16, 11, 7, 5, 3, 2, 1, 1, 209, 52, 24, 15, 11, 7, 5, 3, 2, 1, 1, 330, 80, 35, 22, 15, 11, 7, 5, 3, 2, 1, 1, 512, 122, 52, 31, 22, 15, 11, 7, 5, 3

Offset: 1

Henry Bottomley, Feb 01 2001

Weakly unimodal means nondecreasing then nonincreasing.

			Rows are {1,1,2,4,8,15,...}, {1,1,2,3,5,8,...}, {1,1,2,3,5,7,...} etc.
As an upper right triangle:
  1,  1,  2,  4,  8, 15, ...,
      1,  1,  2,  3,  5,  8, ...,
          1,  1,  2,  3,  5,  7, ...,
              ...
As a left downward triangle, it starts:
   1;
   1, 1;
   2, 1, 1;
   4, 2, 1, 1;
   8, 3, 2, 1, 1;
  15, 5, 3, 2, 1, 1;
  27, 8, 5, 3, 2, 1, 1;
  ...
T(9,3)=11 since 9 can be written as 3+6, 3+5+1, 3+4+2, 3+4+1+1, 3+3+3, 3+3+2+1, 3+3+1+1+1, 3+2+2+2, 3+2+2+1+1, 3+2+1+1+1+1 or 3+1+1+1+1+1.

Column sums give A001523. Cf. A008284, A026836, A008284, A059607, A059619.

T(n, k) = S(n, k) - S(n-k, k) + Sum_j[T(n-k, j)] for j >= k, where S(n, k) = A008284(n, k) = Sum_j[S(n-k, j)] for n>k >= j [note reversal] with S[n, n] = 1.

cp's OEIS Frontend

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

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