%I #20 Apr 21 2021 11:25:22

%S 1,1,1,2,1,2,6,2,1,20,8,2,60,2,10,450,1,2,726,2,140,3321,14,2,5896,

%T 572,16,26426,264,2,394406,2,1,226020,20,51886,961584,2,22,2044895,

%U 38740,2,20959503,2,676,478164163,26,2,56849086,31201,652968,184947044,980,2,1273706934,6620376,153366,1803937344

%N Number of compositions (ordered partitions) of n into odd divisors of n.

%H Alois P. Heinz, <a href="/A284466/b284466.txt">Table of n, a(n) for n = 0..2000</a>

%H <a href="/index/Com#comp">Index entries for sequences related to compositions</a>

%F a(n) = [x^n] 1/(1 - Sum_{d|n, d positive odd} x^d).

%F a(n) = 1 if n is a power of 2.

%F a(n) = 2 if n is an odd prime.

%e a(10) = 8 because 10 has 4 divisors {1, 2, 5, 10} among which 2 are odd {1, 5} therefore we have [5, 5], [5, 1, 1, 1, 1, 1], [1, 5, 1, 1, 1, 1], [1, 1, 5, 1, 1, 1], [1, 1, 1, 5, 1, 1], [1, 1, 1, 1, 5, 1], [1, 1, 1, 1, 1, 5] and [1, 1, 1, 1, 1, 1, 1, 1, 1, 1].

%p with(numtheory):

%p a:= proc(n) option remember; local b, l;

%p l, b:= select(x-> is(x:: odd), divisors(n)),

%p proc(m) option remember; `if`(m=0, 1,

%p add(`if`(j>m, 0, b(m-j)), j=l))

%p end; b(n)

%p end:

%p seq(a(n), n=0..60); # _Alois P. Heinz_, Mar 30 2017

%t Table[d = Divisors[n]; Coefficient[Series[1/(1 - Sum[Boole[Mod[d[[k]], 2] == 1] x^d[[k]], {k, Length[d]}]), {x, 0, n}], x, n], {n, 0, 57}]

%o (Python)

%o from sympy import divisors

%o from sympy.core.cache import cacheit

%o @cacheit

%o def a(n):

%o l=[x for x in divisors(n) if x%2]

%o @cacheit

%o def b(m): return 1 if m==0 else sum(b(m - j) for j in l if j <= m)

%o return b(n)

%o print([a(n) for n in range(61)]) # _Indranil Ghosh_, Aug 01 2017, after Maple code

%Y Cf. A000045, A005408, A032021, A100346, A171565.

%K nonn

%O 0,4

%A _Ilya Gutkovskiy_, Mar 27 2017