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A281489 Number of partitions of n^2 into distinct odd parts. 5
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 (list; graph; refs; listen; history; text; internal format)
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-Fran├žois 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

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Last modified November 28 04:31 EST 2021. Contains 349400 sequences. (Running on oeis4.)