cp's OEIS Frontend

A238628 Number of partitions p of n such that n - max(p) is a part of p.

%I A238628 #16 Oct 12 2023 14:10:31
%S A238628 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 A238628 148,14,190,15,246,16,313,17,402,18,508,19,646,20,812,21,1023,22,1277,
%U A238628 23,1598,24,1982,25,2461,26,3036,27,3745,28,4593,29,5633
%N A238628 Number of partitions p of n such that n - max(p) is a part of p.
%C A238628 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 A238628 a(6) counts these partitions:  51, 42, 33, 321, 3111.
%t A238628 Table[Count[IntegerPartitions[n], p_ /; MemberQ[p, n - Max[p]]], {n, 50}]
%o A238628 (Python)
%o A238628 from sympy.utilities.iterables import partitions
%o A238628 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 A238628 (PARI) a(n) = my(res = floor(n/2)); if(!bitand(n, 1), res+=(numbpart(n/2)-1)); res
%Y A238628 Cf. A238479.
%Y A238628 The strict case is A365659, complement A365826.
%Y A238628 The complement is counted by A365825.
%Y A238628 These partitions are ranked by A366318.
%Y A238628 A000041 counts integer partitions, strict A000009.
%Y A238628 A140106 counts strict partitions of length 2, complement A365827.
%Y A238628 A182616 counts partitions of 2n that do not contain n, strict A365828.
%Y A238628 Cf. A004526, A005408, A008967, A035363, A058984, A068911, A086543.
%K A238628 nonn,easy
%O A238628 1,4
%A A238628 _Clark Kimberling_, Mar 02 2014