A296122 - OEIS

%I #15 May 19 2021 09:29:18

%S 1,1,2,5,10,20,40,77,157,285,552,1018,1921,3484,6436,11622,21082,

%T 37550,67681,119318,211792,372003,653496,1137185,1986234,3429650,

%U 5935970,10205907,17537684,29958671,51189932,86967755,147759421,249850696,422123392,710495901

%N Number of twice-partitions of n with no repeated partitions.

%C a(n) is the number of sequences of distinct integer partitions whose sums are weakly decreasing and add up to n.

%H Alois P. Heinz, <a href="/A296122/b296122.txt">Table of n, a(n) for n = 0..1000</a>

%e The a(4) = 10 twice-partitions: (4), (31), (22), (211), (1111), (3)(1), (21)(1), (111)(1), (2)(11), (11)(2).

%p b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0, add(j!*

%p       binomial(combinat[numbpart](i), j)*b(n-i*j, i-1), j=0..n/i)))

%p     end:

%p a:= n-> b(n$2):

%p seq(a(n), n=0..40);  # _Alois P. Heinz_, Dec 06 2017

%t Table[Length[Join@@Table[Select[Tuples[IntegerPartitions/@p],UnsameQ@@#&],{p,IntegerPartitions[n]}]],{n,15}]

%t (* Second program: *)

%t b[n_, i_] := b[n, i] = If[n == 0, 1, If[i < 1, 0, Sum[j!*

%t      Binomial[PartitionsP[i], j]*b[n - i*j, i - 1], {j, 0, n/i}]]];

%t a[n_] := b[n, n];

%t a /@ Range[0, 40] (* _Jean-François Alcover_, May 19 2021, after _Alois P. Heinz_ *)

%Y Cf. A000009, A000041, A047968, A063834, A261049, A273873, A279375, A295279, A296121.

%K nonn

%O 0,3

%A _Gus Wiseman_, Dec 05 2017

%E a(15)-a(34) from _Robert G. Wilson v_, Dec 06 2017