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

Author

Gus Wiseman, Mar 02 2018

Keywords

• nonn

Examples

			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).
		

Links

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

Crossrefs

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

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

Programs

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}]

Formula

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