A281489 Number of partitions of n^2 into distinct odd parts.

1, 1, 1, 2, 5, 12, 33, 93, 276, 833, 2574, 8057, 25565, 81889, 264703, 861889, 2824974, 9311875, 30851395, 102676439, 343112116, 1150785092, 3872588051, 13071583810, 44245023261, 150145281903, 510721124972, 1741020966255, 5947081503460, 20352707950277

Offset: 0

Author

Alois P. Heinz, Jan 22 2017

Keywords

• nonn

Examples

			a(0) = 1: [], the empty partition.
a(1) = 1: [1].
a(2) = 1: [1,3].
a(3) = 2: [1,3,5], [9].
a(4) = 5: [1,3,5,7], [7,9], [5,11], [3,13], [1,15].
a(5) = 12: [1,3,5,7,9], [5,9,11], [5,7,13], [3,9,13], [1,11,13], [3,7,15], [1,9,15], [3,5,17], [1,7,17], [1,5,19], [1,3,21], [25].
		

Links

• Chai Wah Wu, Table of n, a(n) for n = 0..507 (terms 0..200 from Alois P. Heinz)

Crossrefs

Cf. A000009, A000290, A000700, A005408, A072243, A107379.

Programs

Maple

with(numtheory):
b:= proc(n) option remember; `if`(n=0, 1, add(b(n-j)*add(d*
      [0, 1, -1, 1][1+irem(d, 4)], d=divisors(j)), j=1..n)/n)
    end:
a:= n-> b(n^2):
seq(a(n), n=0..30);

Mathematica

b[n_] := b[n] = If[n==0, 1, Sum[b[n-j]*Sum[d*{0, 1, -1, 1}[[1+Mod[d, 4]]], {d, Divisors[j]}], {j, 1, n}]/n];
a[n_] := b[n^2];
Table[a[n], {n, 0, 30}] (* Jean-François Alcover, Mar 23 2017, translated from Maple *)

Formula

a(n) = [x^(n^2)] Product_{j>=0} (1 + x^(2*j+1)).

a(n) = A000700(A000290(n)).

a(n) ~ exp(Pi*n/sqrt(6)) / (2^(7/4) * 3^(1/4) * n^(3/2)). - Vaclav Kotesovec, Apr 10 2017