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Number of integer partitions whose sum plus co-rank is n, where co-rank is maximum of length and largest part.
7

%I #7 Apr 12 2019 08:26:40

%S 1,0,1,0,2,1,3,2,5,5,8,8,14,14,22,24,35,39,54,62,84,97,127,148,192,

%T 224,284,334,418,492,610,716,880,1035,1259,1480,1790,2100,2522,2958,

%U 3533,4135,4916,5742,6798,7928,9344,10878,12778,14846,17378,20156,23520

%N Number of integer partitions whose sum plus co-rank is n, where co-rank is maximum of length and largest part.

%H FindStat, <a href="http://www.findstat.org/StatisticsDatabase/St000784">St000784: The maximum of the length and the largest part of the integer partition</a>

%e The a(4) = 2 through a(12) = 14 partitions:

%e (2) (21) (3) (31) (4) (33) (5) (43) (6)

%e (11) (22) (211) (32) (41) (42) (51) (44)

%e (111) (221) (222) (322) (332) (52)

%e (311) (321) (331) (421) (333)

%e (1111) (2111) (411) (2221) (422)

%e (2211) (3211) (431)

%e (3111) (4111) (511)

%e (11111) (21111) (2222)

%e (3221)

%e (3311)

%e (4211)

%e (22111)

%e (31111)

%e (111111)

%t Table[Sum[Length[Select[IntegerPartitions[k],Max[Length[#],Max[#]]==n-k&]],{k,0,n}],{n,0,30}]

%Y Row sums of A325194.

%Y Cf. A051924, A065770, A071724, A096771, A115720, A252464, A257990, A263297, A325178, A325189, A325192.

%K nonn

%O 0,5

%A _Gus Wiseman_, Apr 12 2019