%I #8 Dec 31 2022 14:53:24

%S 1,1,1,3,4,7,13,20,32,51,83,130,206,320,496,759,1171,1786,2714,4104,

%T 6193,9286,13920,20737,30865,45721,67632,99683,146604,214865,314782,

%U 459136,668867,972425,1410458,2040894,2950839,4253713,6123836,8801349,12627079

%N Number of twice-partitions of n into distinct strict partitions.

%C A twice-partition of n (A063834) is a sequence of integer partitions, one of each part of an integer partition of n.

%H Andrew Howroyd, <a href="/A358914/b358914.txt">Table of n, a(n) for n = 0..100</a>

%e The a(1) = 1 through a(6) = 13 twice-partitions:

%e ((1)) ((2)) ((3)) ((4)) ((5)) ((6))

%e ((21)) ((31)) ((32)) ((42))

%e ((2)(1)) ((3)(1)) ((41)) ((51))

%e ((21)(1)) ((3)(2)) ((321))

%e ((4)(1)) ((4)(2))

%e ((21)(2)) ((5)(1))

%e ((31)(1)) ((21)(3))

%e ((31)(2))

%e ((3)(21))

%e ((32)(1))

%e ((41)(1))

%e ((3)(2)(1))

%e ((21)(2)(1))

%t twiptn[n_]:=Join@@Table[Tuples[IntegerPartitions/@ptn],{ptn,IntegerPartitions[n]}];

%t Table[Length[Select[twiptn[n],UnsameQ@@#&&And@@UnsameQ@@@#&]],{n,0,10}]

%o (PARI) seq(n,k)={my(u=Vec(eta(x^2 + O(x*x^n))/eta(x + O(x*x^n))-1)); Vec(prod(k=1, n, my(c=u[k]); sum(j=0, min(c,n\k), x^(j*k)*c!/(c-j)!, O(x*x^n))))} \\ _Andrew Howroyd_, Dec 31 2022

%Y The unordered version is A050342, non-strict A261049.

%Y This is the distinct case of A270995.

%Y The case of strictly decreasing sums is A279785.

%Y The case of constant sums is A279791.

%Y For distinct instead of weakly decreasing sums we have A336343.

%Y This is the twice-partition case of A358913.

%Y A001970 counts multiset partitions of integer partitions.

%Y A055887 counts sequences of partitions.

%Y A063834 counts twice-partitions.

%Y A330462 counts set systems by total sum and length.

%Y A358830 counts twice-partitions with distinct lengths.

%Y Cf. A000009, A000219, A075900, A271619, A296122, A304969, A321449, A336342, A358901, A358906, A358907.

%K nonn

%O 0,4

%A _Gus Wiseman_, Dec 11 2022

%E Terms a(26) and beyond from _Andrew Howroyd_, Dec 31 2022