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A296122
Number of twice-partitions of n with no repeated partitions.
26
1, 1, 2, 5, 10, 20, 40, 77, 157, 285, 552, 1018, 1921, 3484, 6436, 11622, 21082, 37550, 67681, 119318, 211792, 372003, 653496, 1137185, 1986234, 3429650, 5935970, 10205907, 17537684, 29958671, 51189932, 86967755, 147759421, 249850696, 422123392, 710495901
OFFSET
0,3
COMMENTS
a(n) is the number of sequences of distinct integer partitions whose sums are weakly decreasing and add up to n.
LINKS
EXAMPLE
The a(4) = 10 twice-partitions: (4), (31), (22), (211), (1111), (3)(1), (21)(1), (111)(1), (2)(11), (11)(2).
MAPLE
b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0, add(j!*
binomial(combinat[numbpart](i), j)*b(n-i*j, i-1), j=0..n/i)))
end:
a:= n-> b(n$2):
seq(a(n), n=0..40); # Alois P. Heinz, Dec 06 2017
MATHEMATICA
Table[Length[Join@@Table[Select[Tuples[IntegerPartitions/@p], UnsameQ@@#&], {p, IntegerPartitions[n]}]], {n, 15}]
(* Second program: *)
b[n_, i_] := b[n, i] = If[n == 0, 1, If[i < 1, 0, Sum[j!*
Binomial[PartitionsP[i], j]*b[n - i*j, i - 1], {j, 0, n/i}]]];
a[n_] := b[n, n];
a /@ Range[0, 40] (* Jean-François Alcover, May 19 2021, after Alois P. Heinz *)
KEYWORD
nonn
AUTHOR
Gus Wiseman, Dec 05 2017
EXTENSIONS
a(15)-a(34) from Robert G. Wilson v, Dec 06 2017
STATUS
approved