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%I #4 Mar 08 2014 22:52:00
%S 0,2,2,5,4,11,9,22,20,42,40,77,77,135,141,231,247,385,420,627,696,
%T 1002,1124,1575,1782,2436,2776,3718,4256,5604,6437,8349,9617,12310,
%U 14203,17977,20764,26015,30070,37338,43166,53174,61469,75175,86879,105558,121926
%N Number of partitions of n such that either both floor(n/2) and ceiling(n/2) are parts or else neither is a part.
%e a(7) counts these 9 partitions: 7, 61, 52, 511, 43, 2221, 22111, 211111, 1111111.
%t z=40; g[n_] := g[n] = IntegerPartitions[n];
%t t1 = Table[Count[g[n], p_ /; Or[MemberQ[p, Floor[n/2]], MemberQ[p, Ceiling[n/2]]]], {n, z}] (* A238622 [or] *)
%t t2 = Table[Count[g[n], p_ /; Nor[MemberQ[p, Floor[n/2]], MemberQ[p, Ceiling[n/2]]]], {n, z}] (* A238623 [nor] *)
%t t3 = Table[Count[g[n], p_ /; Xnor[MemberQ[p, Floor[n/2]], MemberQ[p, Ceiling[n/2]]]], {n, z}] (* A238624 [xnor] *)
%Y Cf. A238622, A238623.
%K nonn,easy
%O 1,2
%A _Clark Kimberling_, Mar 02 2014