|
|
A347446
|
|
Number of integer partitions of n with integer alternating product.
|
|
28
|
|
|
1, 1, 2, 3, 5, 6, 10, 12, 18, 22, 31, 37, 54, 62, 84, 100, 134, 157, 207, 241, 314, 363, 463, 537, 685, 785, 985, 1138, 1410, 1616, 1996, 2286, 2801, 3201, 3885, 4434, 5363, 6098, 7323, 8329, 9954, 11293, 13430, 15214, 18022, 20383, 24017, 27141, 31893, 35960
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
OFFSET
|
0,3
|
|
COMMENTS
|
We define the alternating product of a sequence (y_1,...,y_k) to be Product_i y_i^((-1)^(i-1)).
|
|
LINKS
|
|
|
EXAMPLE
|
The a(1) = 1 through a(7) = 12 partitions:
(1) (2) (3) (4) (5) (6) (7)
(11) (21) (22) (41) (33) (61)
(111) (31) (221) (42) (322)
(211) (311) (51) (331)
(1111) (2111) (222) (421)
(11111) (411) (511)
(2211) (2221)
(3111) (4111)
(21111) (22111)
(111111) (31111)
(211111)
(1111111)
|
|
MATHEMATICA
|
altprod[q_]:=Product[q[[i]]^(-1)^(i-1), {i, Length[q]}];
Table[Length[Select[IntegerPartitions[n], IntegerQ[altprod[#]]&]], {n, 0, 30}]
|
|
CROSSREFS
|
Allowing any reverse-alternating product >= 1 gives A344607.
Allowing any reverse-alternating product < 1 gives A344608.
The multiplicative version (factorizations) is A347437, reverse A347442.
A027187 counts partitions of even length.
A027193 counts partitions of odd length.
A103919 counts partitions by sum and alternating sum (reverse: A344612).
A347461 counts possible alternating products of partitions.
Cf. A025047, A067661, A339890, A347450, A344654, A344740, A347439, A347440, A347451, A347463, A347705.
|
|
KEYWORD
|
nonn
|
|
AUTHOR
|
|
|
STATUS
|
approved
|
|
|
|