A300300 Number of ways to choose a multiset of strict partitions, or odd partitions, of odd numbers, whose weights sum to n.

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

Offset

0,4

Links

Alois P. Heinz, Table of n, a(n) for n = 0..10000

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:
seq(a(n), n=0..45);  # Alois P. Heinz, Mar 02 2018

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.

Sequence in context: A058628 A035528 A341241 * A293675 A050337 A299473

Adjacent sequences: A300297 A300298 A300299 * A300301 A300302 A300303

Keyword

nonn

Author

Gus Wiseman, Mar 02 2018

Status

approved