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A363132 Number of integer partitions of 2n such that 2*(minimum) = (mean).

%I A363132 #15 Dec 30 2023 21:24:01
%S A363132 0,0,1,2,5,6,15,14,32,34,65,55,150,100,225,237,425,296,824,489,1267,
%T A363132 1133,1809,1254,4018,2142,4499,4550,7939,4564,14571,6841,18285,16047,
%U A363132 23408,17495,52545,21636,49943,51182,92516,44582,144872,63260,175318,169232,205353
%N A363132 Number of integer partitions of 2n such that 2*(minimum) = (mean).
%C A363132 Equivalently, n = (length)*(minimum).
%e A363132 The a(2) = 1 through a(7) = 14 partitions:
%e A363132   (31)  (321)  (62)    (32221)  (93)      (3222221)
%e A363132         (411)  (3221)  (33211)  (552)     (3322211)
%e A363132                (3311)  (42211)  (642)     (3332111)
%e A363132                (4211)  (43111)  (732)     (4222211)
%e A363132                (5111)  (52111)  (822)     (4322111)
%e A363132                        (61111)  (322221)  (4331111)
%e A363132                                 (332211)  (4421111)
%e A363132                                 (333111)  (5222111)
%e A363132                                 (422211)  (5321111)
%e A363132                                 (432111)  (5411111)
%e A363132                                 (441111)  (6221111)
%e A363132                                 (522111)  (6311111)
%e A363132                                 (531111)  (7211111)
%e A363132                                 (621111)  (8111111)
%e A363132                                 (711111)
%t A363132 Table[Length[Select[IntegerPartitions[2n],2*Min@@#==Mean[#]&]],{n,0,15}]
%o A363132 (Python)
%o A363132 from sympy.utilities.iterables import partitions
%o A363132 def A363132(n): return sum(1 for s,p in partitions(n<<1,m=n,size=True) if n==s*min(p,default=0)) if n else 0 # _Chai Wah Wu_, Sep 21 2023
%Y A363132 Removing the factor 2 gives A099777.
%Y A363132 Taking maximum instead of mean and including odd indices gives A118096.
%Y A363132 For length instead of mean and including odd indices we have A237757.
%Y A363132 For (maximum) = 2*(mean) see A361851, A361852, A361853, A361854, A361855.
%Y A363132 For median instead of mean we have A361861.
%Y A363132 These partitions have ranks A363133.
%Y A363132 For maximum instead of minimum we have A363218.
%Y A363132 For median instead of minimum we have A363224.
%Y A363132 A000041 counts integer partitions, strict A000009.
%Y A363132 A008284 counts partitions by length, A058398 by mean.
%Y A363132 A051293 counts subsets with integer mean.
%Y A363132 A067538 counts partitions with integer mean.
%Y A363132 A268192 counts partitions by complement size, ranks A326844.
%Y A363132 Cf. A053263, A111907, A237753, A237755, A237824, A327482, A349156, A361906, A361907, A363134.
%K A363132 nonn
%O A363132 0,4
%A A363132 _Gus Wiseman_, May 23 2023
%E A363132 a(31)-a(46) from _Chai Wah Wu_, Sep 21 2023