login
The OEIS Foundation is supported by donations from users of the OEIS and by a grant from the Simons Foundation.

 

Logo


Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A072706 Number of unimodal partitions/compositions of n into distinct terms. 38
1, 1, 1, 3, 3, 5, 9, 11, 15, 21, 33, 39, 55, 69, 93, 127, 159, 201, 261, 327, 411, 537, 653, 819, 1011, 1257, 1529, 1899, 2331, 2829, 3441, 4179, 5031, 6093, 7305, 8767, 10575, 12573, 14997, 17847, 21223, 25089, 29757, 35055, 41379, 48801, 57285, 67131 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,4

COMMENTS

Also the number of ways to partition a strict integer partition of n into two unordered blocks. - Gus Wiseman, Dec 31 2019

LINKS

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

FORMULA

a(n) = sum_k A072705(n, k) = A032020(n)-A072707(k) = A032302(n)/2 (n>0).

G.f.: 1/2*(1+Product_{k>0} (1+2*x^k)). - Vladeta Jovovic, Jun 24 2003

G.f.: 1 + sum(n>=1, 2^(n-1)*q^(n*(n+1)/2) / prod(k=1..n, 1-q^k ) ). [Joerg Arndt, Jan 20 2014]

a(n) ~ c^(1/4) * exp(2*sqrt(c*n)) / (4*sqrt(3*Pi)*n^(3/4)), where c = -polylog(2, -2) = A266576 = 1.436746366883680946362902023893583354... - Vaclav Kotesovec, Sep 22 2019

EXAMPLE

a(6)=9 since 6 can be written as 1+2+3, 1+3+2, 1+5, 2+3+1, 2+4, 3+2+1, 4+2, 5+1, or 6, but not for example 1+4+1 (which does not have distinct terms) nor 2+1+3 (which is not unimodal).

From Joerg Arndt, Mar 25 2014: (Start)

The a(10) = 33 such compositions of 10 are:

01:  [ 1 2 3 4 ]

02:  [ 1 2 4 3 ]

03:  [ 1 2 7 ]

04:  [ 1 3 4 2 ]

05:  [ 1 3 6 ]

06:  [ 1 4 3 2 ]

07:  [ 1 4 5 ]

08:  [ 1 5 4 ]

09:  [ 1 6 3 ]

10:  [ 1 7 2 ]

11:  [ 1 9 ]

12:  [ 2 3 4 1 ]

13:  [ 2 3 5 ]

14:  [ 2 4 3 1 ]

15:  [ 2 5 3 ]

16:  [ 2 7 1 ]

17:  [ 2 8 ]

18:  [ 3 4 2 1 ]

19:  [ 3 5 2 ]

20:  [ 3 6 1 ]

21:  [ 3 7 ]

22:  [ 4 3 2 1 ]

23:  [ 4 5 1 ]

24:  [ 4 6 ]

25:  [ 5 3 2 ]

26:  [ 5 4 1 ]

27:  [ 6 3 1 ]

28:  [ 6 4 ]

29:  [ 7 2 1 ]

30:  [ 7 3 ]

31:  [ 8 2 ]

32:  [ 9 1 ]

33:  [ 10 ]

(End)

MAPLE

b:= proc(n, i) option remember; `if`(n>i*(i+1)/2, 0, `if`(n=0, 1,

      expand(b(n, i-1) +`if`(i>n, 0, x*b(n-i, i-1)))))

    end:

a:= n->(p->add(coeff(p, x, i)*ceil(2^(i-1)), i=0..degree(p)))(b(n$2)):

seq(a(n), n=0..100);  # Alois P. Heinz, Mar 25 2014

MATHEMATICA

b[n_, i_] := b[n, i] = If[n > i*(i + 1)/2, 0, If[n == 0, 1, Expand[b[n, i - 1] + If[i > n, 0, x*b[n - i, i - 1]]]]]; a[n_] := Function[{p}, Sum[Coefficient[p, x, i]*Ceiling[2^(i - 1)], {i, 0, Exponent[p, x]}]][b[n, n]]; Table[a[n], {n, 0, 100}] (* Jean-Fran├žois Alcover, Jan 16 2015, after Alois P. Heinz *)

Table[If[n==0, 1, Sum[2^(Length[ptn]-1), {ptn, Select[IntegerPartitions[n], UnsameQ@@#&]}]], {n, 0, 15}] (* Gus Wiseman, Dec 31 2019 *)

PROG

(PARI) N=66; q='q+O('q^N); Vec( 1 + sum(n=1, N, 2^(n-1)*q^(n*(n+1)/2) / prod(k=1, n, 1-q^k ) ) ) \\ Joerg Arndt, Mar 25 2014

CROSSREFS

The non-strict version is A001523.

Cf. A000009, A000041, A001970, A032020, A059618, A072705, A072707, A270995, A294617.

Sequence in context: A091916 A102437 A319794 * A117433 A159284 A078028

Adjacent sequences:  A072703 A072704 A072705 * A072707 A072708 A072709

KEYWORD

nonn

AUTHOR

Henry Bottomley, Jul 04 2002

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
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified January 26 10:14 EST 2021. Contains 340435 sequences. (Running on oeis4.)