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A052816
Expansion of g.f. (1+x)*Product_{m>0} (1 + x^m).
5
1, 2, 2, 3, 4, 5, 7, 9, 11, 14, 18, 22, 27, 33, 40, 49, 59, 70, 84, 100, 118, 140, 165, 193, 226, 264, 307, 357, 414, 478, 552, 636, 730, 838, 960, 1097, 1253, 1428, 1624, 1846, 2095, 2373, 2686, 3036, 3426, 3864, 4352, 4894, 5500, 6174, 6922, 7755, 8679, 9702
OFFSET
0,2
COMMENTS
Number of partitions of n into distinct parts where there are 2 sorts of ones. Also number of partitions of n where all parts except possibly the largest are odd, see example. [Joerg Arndt, Jun 09 2013]
FORMULA
G.f.: exp( sum(j>=1, (-1)^(j+1) * x^j * (x^j-2)/(x^j-1)/j ) ).
a(n) = A000009(n) + A000009(n-1). - Vladeta Jovovic, Jun 23 2003
a(n) ~ exp(sqrt(n/3)*Pi) / (2*3^(1/4)*n^(3/4)) * (1 - (3*sqrt(3)/(8*Pi) + 11*Pi/(48*sqrt(3)))/sqrt(n) + (55/128 - 45/(128*Pi^2) + 265*Pi^2/13824)/n). - Vaclav Kotesovec, Nov 04 2016
G.f.: A(x) = 1/2 * Sum_{n >= 0} x^((n-1)*(n-2)/2) / (Product_{k = 1..n} 1 - x^k). - Peter Bala, Feb 03 2025
EXAMPLE
From Joerg Arndt, Jun 09 2013: (Start)
There are a(12)=27 partitions of 12 where all parts except possibly the largest are odd:
01: [ 1 1 1 1 1 1 1 1 1 1 1 1 ]
02: [ 1 1 1 1 1 1 1 1 1 1 2 ]
03: [ 1 1 1 1 1 1 1 1 1 3 ]
04: [ 1 1 1 1 1 1 1 1 4 ]
05: [ 1 1 1 1 1 1 1 5 ]
06: [ 1 1 1 1 1 1 3 3 ]
07: [ 1 1 1 1 1 1 6 ]
08: [ 1 1 1 1 1 3 4 ]
09: [ 1 1 1 1 1 7 ]
10: [ 1 1 1 1 3 5 ]
11: [ 1 1 1 1 8 ]
12: [ 1 1 1 3 3 3 ]
13: [ 1 1 1 3 6 ]
14: [ 1 1 1 9 ]
15: [ 1 1 3 3 4 ]
16: [ 1 1 3 7 ]
17: [ 1 1 5 5 ]
18: [ 1 1 10 ]
19: [ 1 3 3 5 ]
20: [ 1 3 8 ]
21: [ 1 5 6 ]
22: [ 1 11 ]
23: [ 3 3 3 3 ]
24: [ 3 3 6 ]
25: [ 3 9 ]
26: [ 5 7 ]
27: [ 12 ]
(End)
MAPLE
spec := [S, {B=Sequence(Z, 1 <= card), C=Union(B, Z), S=PowerSet(C)}, unlabeled]: seq(combstruct[count](spec, size=n), n=0..20);
# Alternative:
series(1/2 * add( x^((n-1)*(n-2)/2) / mul(1 - x^k, k = 1..n), n = 0..11), x, 51):
seq(coeftayl(%, x = 0, n), n = 0..50); # Peter Bala, Feb 03 2025
MATHEMATICA
a[n_] := PartitionsQ[n] + PartitionsQ[n-1];
Table[a[n], {n, 0, 100}] (* Jean-François Alcover, Oct 11 2022, after Vladeta Jovovic *)
CROSSREFS
Sequence in context: A304883 A393565 A280663 * A122130 A003073 A123946
KEYWORD
easy,nonn
EXTENSIONS
More terms from Vladeta Jovovic, Jun 23 2003
STATUS
approved