A300301 Number of ways to choose a partition, with odd parts, of each part of a partition of n into odd parts.

1, 1, 1, 3, 3, 6, 10, 15, 21, 37, 56, 80, 127, 183, 280, 428, 616, 893, 1367, 1944, 2846, 4223, 6049, 8691, 12670, 18128, 25921, 37529, 53338, 75738, 108561, 153460, 216762, 308829, 433893, 612006, 864990, 1211097, 1697020, 2386016, 3331037, 4648229, 6503314

Offset

0,4

Links

Alois P. Heinz, Table of n, a(n) for n = 0..5000

Formula

O.g.f.: Product_{n odd} 1/(1 - A000009(n)x^n).

Example

The a(6) = 10 twice-partitions using odd partitions: (5)(1), (3)(3), (113)(1), (3)(111), (111)(3), (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(add(
     `if`(d::odd, d, 0), d=divisors(j))*b(n-j), j=1..n)/n)
    end:
g:= proc(n, i) option remember; `if`(n=0 or i=1, 1,
      g(n, i-2)+`if`(i>n, 0, b(i)*g(n-i, i)))
    end:
a:= n-> g(n, n-1+irem(n, 2)):
seq(a(n), n=0..50);  # Alois P. Heinz, Mar 05 2018

Mathematica

nn=50;
ser=Product[1/(1-PartitionsQ[n]x^n), {n, 1, nn, 2}];
Table[SeriesCoefficient[ser, {x, 0, n}], {n, 0, nn}]

Crossrefs

A300300(n) <= a(n) <= A279785(n)

Cf. A000009, A063834, A078408, A089259, A270995, A271619, A279374, A279375, A279790, A294617, A300300.

Sequence in context: A266137 A265506 A366317 * A031504 A298164 A304265

Adjacent sequences: A300298 A300299 A300300 * A300302 A300303 A300304

Keyword

nonn

Author

Gus Wiseman, Mar 02 2018

Status

approved