%I #8 Jan 28 2022 01:08:42
%S 0,0,0,0,1,1,1,0,1,2,2,3,2,3,4,5,6,7,8,10,12,14,16,20,23,27,32,37,43,
%T 52,59,69,80,93,108,126,144,166,192,222,254,294,334,384,441,504,575,
%U 658,748,854,972,1106,1254,1428,1617,1834,2077,2350,2656,3007
%N Number of (4,1)-separable partitions of n; see Comments.
%C Suppose that p is a partition of n into 2 or more parts and that h is a part of p. Then p is (h,0)-separable if there is an ordering x, h, x, h, ..., h, x of the parts of p, where each x represents any part of p except h. Here, the number of h's on the ends of the ordering is 0. Similarly, p is (h,1)-separable if there is an ordering x, h, x, h, ..., x, h, where the number of h's on the ends is 1; next, p is (h,2)-separable if there is an ordering h, x, h, ..., x, h. Finally, p is h-separable if it is (h,i)-separable for i = 0,1,2.
%e The (4,1)-separable partitions of 12 are 84, 3414, 2424, so that a(12) = 3.
%t z = 70; Table[Count[IntegerPartitions[n], p_ /; 2 Count[p, 1] == Length[p]], {n, 1, z}] (* A008483 *)
%t Table[Count[IntegerPartitions[n], p_ /; 2 Count[p, 2] == Length[p]], {n, 1, z}] (* A239493 *)
%t Table[Count[IntegerPartitions[n], p_ /; 2 Count[p, 3] == Length[p]], {n, 1, z}] (* A239494 *)
%t Table[Count[IntegerPartitions[n], p_ /; 2 Count[p, 4] == Length[p]], {n, 1, z}] (* A239495 *)
%t Table[Count[IntegerPartitions[n], p_ /; 2 Count[p, 5] == Length[p]], {n, 1, z}] (* A239496 *)
%Y Cf. A230467, A008483, A239493, A239494, A239496.
%K nonn,easy
%O 1,10
%A _Clark Kimberling_, Mar 20 2014
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