A279374 Number of ways to choose an odd partition of each part of an odd partition of 2n+1.

1, 3, 6, 15, 37, 80, 183, 428, 893, 1944, 4223, 8691, 18128, 37529, 75738, 153460, 308829, 612006, 1211097, 2386016, 4648229, 9042678, 17528035, 33645928, 64508161, 123178953, 233709589, 442583046, 834923483, 1567271495, 2935406996, 5481361193, 10191781534

Offset: 0

Author

Gus Wiseman, Dec 11 2016

Keywords

• nonn

Comments

An odd partition is an integer partition of an odd number with an odd number of parts, all of which are odd.

Examples

			The a(3)=15 ways to choose an odd partition of each part of an odd partition of 7 are:
((7)), ((511)), ((331)), ((31111)), ((1111111)), ((5)(1)(1)), ((311)(1)(1)),
((11111)(1)(1)), ((3)(3)(1)), ((3)(111)(1)), ((111)(3)(1)), ((111)(111)(1)),
((3)(1)(1)(1)(1)), ((111)(1)(1)(1)(1)), ((1)(1)(1)(1)(1)(1)(1)).
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..4919
• Gus Wiseman, "Twice-odd partitions of n=9."

Crossrefs

Cf. A000009 (strict partitions), A078408 (odd partitions), A063834, A271619, A279375.

Programs

Maple

g:= proc(n) option remember; `if`(n=0, 1, add(add(d*
      [0, 2, 0, 1$4, 2, 0, 2, 1$4, 0, 2][1+irem(d, 16)],
      d=numtheory[divisors](j))*g(n-j), j=1..n)/n)
    end:
b:= proc(n, i, t) option remember;
      `if`(n=0, t, `if`(i<1, 0, b(n, i-2, t)+
      `if`(i>n, 0, b(n-i, i, 1-t)*g((i-1)/2))))
    end:
a:= n-> b(2*n+1$2, 0):
seq(a(n), n=0..35);  # Alois P. Heinz, Dec 12 2016

Mathematica

nn=20;Table[SeriesCoefficient[Product[1/(1-PartitionsQ[k]x^k),{k,1,2n-1,2}],{x,0,2n-1}],{n,nn}]