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

0,4

Links

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

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

Example

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].

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 *)

Crossrefs

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

Sequence in context: A295461 A191769 A221206 * A225616 A186739 A266292

Adjacent sequences: A281486 A281487 A281488 * A281490 A281491 A281492

Keyword

nonn

Author

Alois P. Heinz, Jan 22 2017

Status

approved