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%I #27 Oct 11 2022 06:07:59
%S 1,2,2,3,4,5,7,9,11,14,18,22,27,33,40,49,59,70,84,100,118,140,165,193,
%T 226,264,307,357,414,478,552,636,730,838,960,1097,1253,1428,1624,1846,
%U 2095,2373,2686,3036,3426,3864,4352,4894,5500,6174,6922,7755,8679,9702
%N G.f.: (1+x)*Product_{m>0} (1 + x^m).
%C 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]
%H INRIA Algorithms Project, <a href="http://ecs.inria.fr/services/structure?nbr=781">Encyclopedia of Combinatorial Structures 781</a>
%F G.f.: exp( sum(j>=1, (-1)^(j+1) * x^j * (x^j-2)/(x^j-1)/j ) ).
%F a(n) = A000009(n) + A000009(n-1). - _Vladeta Jovovic_, Jun 23 2003
%F 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
%e From _Joerg Arndt_, Jun 09 2013: (Start)
%e There are a(12)=27 partitions of 12 where all parts except possibly the largest are odd:
%e 01: [ 1 1 1 1 1 1 1 1 1 1 1 1 ]
%e 02: [ 1 1 1 1 1 1 1 1 1 1 2 ]
%e 03: [ 1 1 1 1 1 1 1 1 1 3 ]
%e 04: [ 1 1 1 1 1 1 1 1 4 ]
%e 05: [ 1 1 1 1 1 1 1 5 ]
%e 06: [ 1 1 1 1 1 1 3 3 ]
%e 07: [ 1 1 1 1 1 1 6 ]
%e 08: [ 1 1 1 1 1 3 4 ]
%e 09: [ 1 1 1 1 1 7 ]
%e 10: [ 1 1 1 1 3 5 ]
%e 11: [ 1 1 1 1 8 ]
%e 12: [ 1 1 1 3 3 3 ]
%e 13: [ 1 1 1 3 6 ]
%e 14: [ 1 1 1 9 ]
%e 15: [ 1 1 3 3 4 ]
%e 16: [ 1 1 3 7 ]
%e 17: [ 1 1 5 5 ]
%e 18: [ 1 1 10 ]
%e 19: [ 1 3 3 5 ]
%e 20: [ 1 3 8 ]
%e 21: [ 1 5 6 ]
%e 22: [ 1 11 ]
%e 23: [ 3 3 3 3 ]
%e 24: [ 3 3 6 ]
%e 25: [ 3 9 ]
%e 26: [ 5 7 ]
%e 27: [ 12 ]
%e (End)
%p spec := [S,{B=Sequence(Z,1 <= card),C=Union(B,Z),S=PowerSet(C)},unlabeled]: seq(combstruct[count](spec,size=n), n=0..20);
%t a[n_] := PartitionsQ[n] + PartitionsQ[n-1];
%t Table[a[n], {n, 0, 100}] (* _Jean-François Alcover_, Oct 11 2022, after _Vladeta Jovovic_ *)
%Y Cf. A000009.
%K easy,nonn
%O 0,2
%A encyclopedia(AT)pommard.inria.fr, Jan 25 2000
%E More terms from _Vladeta Jovovic_, Jun 23 2003
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