|
|
A300300
|
|
Number of ways to choose a multiset of strict partitions, or odd partitions, of odd numbers, whose weights sum to n.
|
|
13
|
|
|
1, 1, 1, 3, 3, 6, 9, 14, 20, 32, 48, 69, 105, 150, 225, 322, 472, 669, 977, 1379, 1980, 2802, 3977, 5602, 7892, 11083, 15494, 21688, 30147, 42007, 58143, 80665, 111199, 153640, 211080, 290408, 397817, 545171, 744645, 1016826, 1385124, 1885022, 2561111, 3474730
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
OFFSET
|
0,4
|
|
LINKS
|
|
|
FORMULA
|
Euler transform of {Q(1), 0, Q(3), 0, Q(5), 0, ...} where Q = A000009.
|
|
EXAMPLE
|
The a(6) = 9 multiset partitions using odd-weight strict partitions: (5)(1), (14)(1), (3)(3), (32)(1), (3)(21), (3)(1)(1)(1), (21)(21), (21)(1)(1)(1), (1)(1)(1)(1)(1)(1).
The a(6) = 9 multiset partitions using odd partitions: (5)(1), (3)(3), (311)(1), (3)(111), (3)(1)(1)(1), (11111)(1), (111)(111), (111)(1)(1)(1), (1)(1)(1)(1)(1)(1).
|
|
MAPLE
|
with(numtheory):
b:= proc(n) option remember; `if`(n=0, 1, add(b(n-j)*add(
`if`(d::odd, d, 0), d=divisors(j)), j=1..n)/n)
end:
a:= proc(n) option remember; `if`(n=0, 1, add(a(n-j)*add(
`if`(d::odd, b(d)*d, 0), d=divisors(j)), j=1..n)/n)
end:
|
|
MATHEMATICA
|
nn=50;
ser=Product[1/(1-x^n)^PartitionsQ[n], {n, 1, nn, 2}];
Table[SeriesCoefficient[ser, {x, 0, n}], {n, 0, nn}]
|
|
CROSSREFS
|
Cf. A000009, A063834, A078408, A089259, A270995, A271619, A279374, A279375, A279785, A279790, A294617, A300301.
|
|
KEYWORD
|
nonn
|
|
AUTHOR
|
|
|
STATUS
|
approved
|
|
|
|