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A327640 Number of partitions of n into divisors d of n such that n/d is odd.

%I A327640 #17 Sep 08 2022 08:46:24
%S A327640 1,1,1,2,1,2,2,2,1,5,2,2,2,2,2,14,1,2,5,2,2,18,2,2,2,7,2,23,2,2,14,2,
%T A327640 1,26,2,26,5,2,2,30,2,2,18,2,2,286,2,2,2,9,7,38,2,2,23,38,2,42,2,2,14,
%U A327640 2,2,493,1,44,26,2,2,50,26,2,5,2,2,698,2,50,30,2,2
%N A327640 Number of partitions of n into divisors d of n such that n/d is odd.
%H A327640 Alois P. Heinz, <a href="/A327640/b327640.txt">Table of n, a(n) for n = 0..20000</a> (first 3001 terms from Michael De Vlieger)
%F A327640 a(n) = [x^n] Product_{d|n, n/d odd} 1 / (1 - x^d).
%F A327640 a(2^k) = 1.
%p A327640 with(numtheory):
%p A327640 a:= proc(n) option remember; local b, l; l, b:= sort(
%p A327640       [select(x-> is((n/x):: odd), divisors(n))[]]),
%p A327640       proc(m, i) option remember; `if`(m=0, 1, `if`(i<1, 0,
%p A327640         b(m, i-1)+`if`(l[i]>m, 0, b(m-l[i], i))))
%p A327640       end; b(n, nops(l))
%p A327640     end:
%p A327640 seq(a(n), n=0..100);  # _Alois P. Heinz_, Dec 06 2021
%t A327640 a[n_] := SeriesCoefficient[Product[1/(1 - Boole[OddQ[n/d]] x^d), {d, Divisors[n]}], {x, 0, n}]; Table[a[n], {n, 0, 80}]
%o A327640 (Magma) [1] cat [#RestrictedPartitions(n,{d:d in Divisors(n)|IsOdd(n div d)}):n in [1..80]]; // _Marius A. Burtea_, Sep 20 2019
%Y A327640 Cf. A018818, A038550 (positions of 2's), A171565.
%K A327640 nonn
%O A327640 0,4
%A A327640 _Ilya Gutkovskiy_, Sep 20 2019