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Number of partitions p of n such that n - max(p) is a part of p.
27

%I #16 Oct 12 2023 14:10:31

%S 0,1,1,3,2,5,3,8,4,11,5,16,6,21,7,29,8,38,9,51,10,66,11,88,12,113,13,

%T 148,14,190,15,246,16,313,17,402,18,508,19,646,20,812,21,1023,22,1277,

%U 23,1598,24,1982,25,2461,26,3036,27,3745,28,4593,29,5633

%N Number of partitions p of n such that n - max(p) is a part of p.

%C Also the number of integer partitions of n that are of length 2 or contain n/2. The first condition alone is A004526, complement A058984. The second condition alone is A035363, complement A086543, ranks A344415. - _Gus Wiseman_, Oct 07 2023

%e a(6) counts these partitions: 51, 42, 33, 321, 3111.

%t Table[Count[IntegerPartitions[n], p_ /; MemberQ[p, n - Max[p]]], {n, 50}]

%o (Python)

%o from sympy.utilities.iterables import partitions

%o def A238628(n): return sum(1 for p in partitions(n) if n-max(p,default=0) in p) # _Chai Wah Wu_, Sep 21 2023

%o (PARI) a(n) = my(res = floor(n/2)); if(!bitand(n, 1), res+=(numbpart(n/2)-1)); res

%Y Cf. A238479.

%Y The strict case is A365659, complement A365826.

%Y The complement is counted by A365825.

%Y These partitions are ranked by A366318.

%Y A000041 counts integer partitions, strict A000009.

%Y A140106 counts strict partitions of length 2, complement A365827.

%Y A182616 counts partitions of 2n that do not contain n, strict A365828.

%Y Cf. A004526, A005408, A008967, A035363, A058984, A068911, A086543.

%K nonn,easy

%O 1,4

%A _Clark Kimberling_, Mar 02 2014