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

0,2

Comments

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

Links

Alois P. Heinz, Table of n, a(n) for n = 0..4919

Gus Wiseman, "Twice-odd partitions of n=9."

Example

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)).

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}]

Crossrefs

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

Sequence in context: A370241 A058534 A063778 * A087124 A327647 A086326

Adjacent sequences: A279371 A279372 A279373 * A279375 A279376 A279377

Keyword

nonn

Author

Gus Wiseman, Dec 11 2016

Status

approved