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

%I A281489 #24 Sep 09 2021 22:26:11
%S A281489 1,1,1,2,5,12,33,93,276,833,2574,8057,25565,81889,264703,861889,
%T A281489 2824974,9311875,30851395,102676439,343112116,1150785092,3872588051,
%U A281489 13071583810,44245023261,150145281903,510721124972,1741020966255,5947081503460,20352707950277
%N A281489 Number of partitions of n^2 into distinct odd parts.
%H A281489 Chai Wah Wu, <a href="/A281489/b281489.txt">Table of n, a(n) for n = 0..507</a> (terms 0..200 from Alois P. Heinz)
%F A281489 a(n) = [x^(n^2)] Product_{j>=0} (1 + x^(2*j+1)).
%F A281489 a(n) = A000700(A000290(n)).
%F A281489 a(n) ~ exp(Pi*n/sqrt(6)) / (2^(7/4) * 3^(1/4) * n^(3/2)). - _Vaclav Kotesovec_, Apr 10 2017
%e A281489 a(0) = 1: [], the empty partition.
%e A281489 a(1) = 1: [1].
%e A281489 a(2) = 1: [1,3].
%e A281489 a(3) = 2: [1,3,5], [9].
%e A281489 a(4) = 5: [1,3,5,7], [7,9], [5,11], [3,13], [1,15].
%e A281489 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].
%p A281489 with(numtheory):
%p A281489 b:= proc(n) option remember; `if`(n=0, 1, add(b(n-j)*add(d*
%p A281489       [0, 1, -1, 1][1+irem(d, 4)], d=divisors(j)), j=1..n)/n)
%p A281489     end:
%p A281489 a:= n-> b(n^2):
%p A281489 seq(a(n), n=0..30);
%t A281489 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];
%t A281489 a[n_] := b[n^2];
%t A281489 Table[a[n], {n, 0, 30}] (* _Jean-François Alcover_, Mar 23 2017, translated from Maple *)
%Y A281489 Cf. A000009, A000290, A000700, A005408, A072243, A107379.
%K A281489 nonn
%O A281489 0,4
%A A281489 _Alois P. Heinz_, Jan 22 2017