cp's OEIS Frontend

A358914 Number of twice-partitions of n into distinct strict partitions.

%I A358914 #8 Dec 31 2022 14:53:24
%S A358914 1,1,1,3,4,7,13,20,32,51,83,130,206,320,496,759,1171,1786,2714,4104,
%T A358914 6193,9286,13920,20737,30865,45721,67632,99683,146604,214865,314782,
%U A358914 459136,668867,972425,1410458,2040894,2950839,4253713,6123836,8801349,12627079
%N A358914 Number of twice-partitions of n into distinct strict partitions.
%C A358914 A twice-partition of n (A063834) is a sequence of integer partitions, one of each part of an integer partition of n.
%H A358914 Andrew Howroyd, <a href="/A358914/b358914.txt">Table of n, a(n) for n = 0..100</a>
%e A358914 The a(1) = 1 through a(6) = 13 twice-partitions:
%e A358914   ((1))  ((2))  ((3))     ((4))      ((5))      ((6))
%e A358914                 ((21))    ((31))     ((32))     ((42))
%e A358914                 ((2)(1))  ((3)(1))   ((41))     ((51))
%e A358914                           ((21)(1))  ((3)(2))   ((321))
%e A358914                                      ((4)(1))   ((4)(2))
%e A358914                                      ((21)(2))  ((5)(1))
%e A358914                                      ((31)(1))  ((21)(3))
%e A358914                                                 ((31)(2))
%e A358914                                                 ((3)(21))
%e A358914                                                 ((32)(1))
%e A358914                                                 ((41)(1))
%e A358914                                                 ((3)(2)(1))
%e A358914                                                 ((21)(2)(1))
%t A358914 twiptn[n_]:=Join@@Table[Tuples[IntegerPartitions/@ptn],{ptn,IntegerPartitions[n]}];
%t A358914 Table[Length[Select[twiptn[n],UnsameQ@@#&&And@@UnsameQ@@@#&]],{n,0,10}]
%o A358914 (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 A358914 The unordered version is A050342, non-strict A261049.
%Y A358914 This is the distinct case of A270995.
%Y A358914 The case of strictly decreasing sums is A279785.
%Y A358914 The case of constant sums is A279791.
%Y A358914 For distinct instead of weakly decreasing sums we have A336343.
%Y A358914 This is the twice-partition case of A358913.
%Y A358914 A001970 counts multiset partitions of integer partitions.
%Y A358914 A055887 counts sequences of partitions.
%Y A358914 A063834 counts twice-partitions.
%Y A358914 A330462 counts set systems by total sum and length.
%Y A358914 A358830 counts twice-partitions with distinct lengths.
%Y A358914 Cf. A000009, A000219, A075900, A271619, A296122, A304969, A321449, A336342, A358901, A358906, A358907.
%K A358914 nonn
%O A358914 0,4
%A A358914 _Gus Wiseman_, Dec 11 2022
%E A358914 Terms a(26) and beyond from _Andrew Howroyd_, Dec 31 2022