login
This site is supported by donations to The OEIS Foundation.

 

Logo


Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A059618 Number of strongly unimodal partitions of n (strongly unimodal means strictly increasing then strictly decreasing). 8
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 (list; graph; refs; listen; history; text; internal format)
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-Fran├žois 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

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent | More pages
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy .

Last modified May 25 10:24 EDT 2017. Contains 287026 sequences.