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Number of integer partitions of n with exactly two distinct parts, both appearing with the same multiplicity.
4

%I #13 Feb 04 2024 21:22:28

%S 0,0,0,1,1,2,3,3,4,5,6,5,9,6,9,10,11,8,15,9,16,14,15,11,23,14,18,18,

%T 23,14,30,15,26,22,24,22,38,18,27,26,38,20,42,21,37,36,33,23,53,27,42,

%U 34,44,26,54,34,53,38,42,29,74,30,45,49,57,40,66,33,58,46

%N Number of integer partitions of n with exactly two distinct parts, both appearing with the same multiplicity.

%C The Heinz numbers of these partitions are given by A268390.

%H Alois P. Heinz, <a href="/A367588/b367588.txt">Table of n, a(n) for n = 0..10000</a>

%F G.f.: Sum_{i, j>0} x^(j*(2*i+1))/(1-x^j). - _John Tyler Rascoe_, Feb 04 2024

%e The a(3) = 1 through a(12) = 9 partitions (A = 10, B = 11):

%e (21) (31) (32) (42) (43) (53) (54) (64) (65) (75)

%e (41) (51) (52) (62) (63) (73) (74) (84)

%e (2211) (61) (71) (72) (82) (83) (93)

%e (3311) (81) (91) (92) (A2)

%e (222111) (3322) (A1) (B1)

%e (4411) (4422)

%e (5511)

%e (333111)

%e (22221111)

%t Table[Sum[Floor[(d-1)/2],{d,Divisors[n]}],{n,30}]

%Y For any multiplicities we have A002133, ranks A007774.

%Y For any number of distinct parts we have A047966, ranks A072774.

%Y For distinct multiplicities we have A182473, ranks A367589.

%Y These partitions have ranks A268390.

%Y A000041 counts integer partitions, strict A000009.

%Y A072233 counts partitions by number of parts.

%Y A116608 counts partitions by number of distinct parts.

%Y Cf. A023645, A091602, A116861, A181819, A243978, A367582.

%K nonn

%O 0,6

%A _Gus Wiseman_, Dec 01 2023