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A072706 Number of unimodal partitions/compositions of n into distinct terms. 5
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

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]

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 *)

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

Cf. A000009, A000041, A001523, A032020, A059618, A072705, A072707.

Sequence in context: A213933 A091916 A102437 * A117433 A159284 A078028

Adjacent sequences:  A072703 A072704 A072705 * A072707 A072708 A072709

KEYWORD

nonn

AUTHOR

Henry Bottomley, Jul 04 2002

STATUS

approved

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Last modified March 27 22:02 EDT 2017. Contains 284182 sequences.