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A072706 Number of unimodal partitions/compositions of n into distinct terms. 42
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
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-François 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.
Sequence in context: A091916 A102437 A319794 * A117433 A349054 A159284
KEYWORD
nonn
AUTHOR
Henry Bottomley, Jul 04 2002
STATUS
approved

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Last modified March 19 01:57 EDT 2024. Contains 370952 sequences. (Running on oeis4.)