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A364686 a(n) is the number of parity self-conjugate partitions of n.

%I A364686 #20 Jan 18 2024 20:21:39
%S A364686 1,0,1,1,1,1,1,4,2,2,2,7,5,3,4,11,11,5,10,17,18,8,17,29,30,16,28,46,
%T A364686 45,28,42,77,69,48,65,119,103,77,97,182,157,118,149,267,236,176,222,
%U A364686 389,353,258,335,551,515,373,494,785,746,534,718,1099,1061,764,1021,1538,1494
%N A364686 a(n) is the number of parity self-conjugate partitions of n.
%C A364686 A partition p is parity self-conjugate if the j-th parts of p and p' have the same parity for every j. If p and p' have different numbers of parts, include terminal 0's as needed.
%C A364686 Such partition p of n has exactly A110654(n) parts for n != 2 and so the largest part is at most A110654(n). - _David A. Corneth_, Dec 09 2023
%F A364686 a(n) >= A000700(n). - _David A. Corneth_, Dec 09 2023
%e A364686 The seven parity self-conjugate partitions of 12 are (6,6), (5, 5, 2), (4, 4, 2, 2), (3, 3, 2, 2, 2), (5, 3, 2, 1, 1), (2, 2, 2, 2, 2, 2), and (6, 2, 1, 1, 1, 1).
%e A364686 From _David A. Corneth_, Dec 09 2023: (Start)
%e A364686 Read as digits these are, with the conjugates, (66, 222222), (552, 33222), (4422, 4422), (33222, 552), (53211, 53211), (22222, 66), (621111, 621111).
%e A364686 66 is extended to 660000 to then check parity of terms in the conjugate 222222.
%e A364686 Note that for example (552, 33222) and (33222, 552) are both counted even though they hold the same partitions, just in a different order. (End)
%t A364686 << "Combinatorica`"
%t A364686 Zs[n_] := Table[0, n]
%t A364686 PadDiff[{L1_, L2_}] := Block[{n1 = Length[L1], n2 = Length[L2]},
%t A364686   Which[n1 < n2, Join[L1, Zs[n2 - n1]] - L2, n1 > n2,
%t A364686    L1 - Join[L2, Zs[n1 - n2]], n1 == n2, L1 - L2 ]]
%t A364686 PSC1[n_] :=
%t A364686  Block[{Pttns = IntegerPartitions[n]},
%t A364686   Union[Flatten[
%t A364686     Select[Transpose[{Pttns, TransposePartition /@ Pttns}],
%t A364686      AllTrue[PadDiff[#], EvenQ] &], 1]]]
%t A364686 Table[Length[PSC1[n]], {n, 1, 50}]
%Y A364686 Cf. A000700, A110654.
%K A364686 nonn
%O A364686 1,8
%A A364686 _Eric Gottlieb_, Aug 02 2023
%E A364686 More terms from _David A. Corneth_, Dec 09 2023