A052816 - OEIS

%I #33 Feb 03 2025 21:28:16

%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 Expansion of 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

%F 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 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);

%p # second program

%p series(1/2 * add( x^((n-1)*(n-2)/2) / mul(1 - x^k, k = 1..n), n = 0..11), x, 51):

%p seq(coeftayl(%, x = 0, n), n = 0..50); # _Peter Bala_, Feb 03 2025

%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, A329289.

%K easy,nonn,changed

%O 0,2

%A encyclopedia(AT)pommard.inria.fr, Jan 25 2000

%E More terms from _Vladeta Jovovic_, Jun 23 2003