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A052816 Expansion of g.f. (1+x)*Product_{m>0} (1 + x^m).

%I A052816 #33 Feb 03 2025 21:28:16
%S A052816 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 A052816 226,264,307,357,414,478,552,636,730,838,960,1097,1253,1428,1624,1846,
%U A052816 2095,2373,2686,3036,3426,3864,4352,4894,5500,6174,6922,7755,8679,9702
%N A052816 Expansion of g.f. (1+x)*Product_{m>0} (1 + x^m).
%C A052816 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 A052816 INRIA Algorithms Project, <a href="http://ecs.inria.fr/services/structure?nbr=781">Encyclopedia of Combinatorial Structures 781</a>
%F A052816 G.f.: exp( sum(j>=1, (-1)^(j+1) * x^j * (x^j-2)/(x^j-1)/j ) ).
%F A052816 a(n) = A000009(n) + A000009(n-1). - _Vladeta Jovovic_, Jun 23 2003
%F A052816 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
%F A052816 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
%e A052816 From _Joerg Arndt_, Jun 09 2013: (Start)
%e A052816 There are a(12)=27 partitions of 12 where all parts except possibly the largest are odd:
%e A052816 01:  [ 1 1 1 1 1 1 1 1 1 1 1 1 ]
%e A052816 02:  [ 1 1 1 1 1 1 1 1 1 1 2 ]
%e A052816 03:  [ 1 1 1 1 1 1 1 1 1 3 ]
%e A052816 04:  [ 1 1 1 1 1 1 1 1 4 ]
%e A052816 05:  [ 1 1 1 1 1 1 1 5 ]
%e A052816 06:  [ 1 1 1 1 1 1 3 3 ]
%e A052816 07:  [ 1 1 1 1 1 1 6 ]
%e A052816 08:  [ 1 1 1 1 1 3 4 ]
%e A052816 09:  [ 1 1 1 1 1 7 ]
%e A052816 10:  [ 1 1 1 1 3 5 ]
%e A052816 11:  [ 1 1 1 1 8 ]
%e A052816 12:  [ 1 1 1 3 3 3 ]
%e A052816 13:  [ 1 1 1 3 6 ]
%e A052816 14:  [ 1 1 1 9 ]
%e A052816 15:  [ 1 1 3 3 4 ]
%e A052816 16:  [ 1 1 3 7 ]
%e A052816 17:  [ 1 1 5 5 ]
%e A052816 18:  [ 1 1 10 ]
%e A052816 19:  [ 1 3 3 5 ]
%e A052816 20:  [ 1 3 8 ]
%e A052816 21:  [ 1 5 6 ]
%e A052816 22:  [ 1 11 ]
%e A052816 23:  [ 3 3 3 3 ]
%e A052816 24:  [ 3 3 6 ]
%e A052816 25:  [ 3 9 ]
%e A052816 26:  [ 5 7 ]
%e A052816 27:  [ 12 ]
%e A052816 (End)
%p A052816 spec := [S,{B=Sequence(Z,1 <= card),C=Union(B,Z),S=PowerSet(C)},unlabeled]: seq(combstruct[count](spec,size=n), n=0..20);
%p A052816 # second program
%p A052816 series(1/2 * add( x^((n-1)*(n-2)/2) / mul(1 - x^k, k = 1..n), n = 0..11), x, 51):
%p A052816 seq(coeftayl(%, x = 0, n), n = 0..50); # _Peter Bala_, Feb 03 2025
%t A052816 a[n_] := PartitionsQ[n] + PartitionsQ[n-1];
%t A052816 Table[a[n], {n, 0, 100}] (* _Jean-François Alcover_, Oct 11 2022, after _Vladeta Jovovic_ *)
%Y A052816 Cf. A000009, A329289.
%K A052816 easy,nonn
%O A052816 0,2
%A A052816 encyclopedia(AT)pommard.inria.fr, Jan 25 2000
%E A052816 More terms from _Vladeta Jovovic_, Jun 23 2003