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A284466 Number of compositions (ordered partitions) of n into odd divisors of n. 2
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, 572, 16, 26426, 264, 2, 394406, 2, 1, 226020, 20, 51886, 961584, 2, 22, 2044895, 38740, 2, 20959503, 2, 676, 478164163, 26, 2, 56849086, 31201, 652968, 184947044, 980, 2, 1273706934, 6620376, 153366, 1803937344 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,4

LINKS

Alois P. Heinz, Table of n, a(n) for n = 0..2000

Index entries for sequences related to compositions

FORMULA

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

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

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

EXAMPLE

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].

MAPLE

with(numtheory):

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

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

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

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

      end; b(n)

    end:

seq(a(n), n=0..60);  # Alois P. Heinz, Mar 30 2017

MATHEMATICA

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}]

PROG

(Python)

from sympy import divisors

class Memoize:

    def __init__(self, func):

        self.func=func

        self.cache={}

    def __call__(self, arg):

        if arg not in self.cache:

            self.cache[arg] = self.func(arg)

        return self.cache[arg]

@Memoize

def a(n):

    l=[x for x in divisors(n) if x%2==1]

    @Memoize

    def b(m): return 1 if m==0 else sum([0 if j>m else b(m - j) for j in l])

    return b(n)

print map(a, xrange(61)) # Indranil Ghosh, Aug 01 2017, after Maple code

CROSSREFS

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

Sequence in context: A265894 A133644 A265870 * A258615 A152431 A143965

Adjacent sequences:  A284463 A284464 A284465 * A284467 A284468 A284469

KEYWORD

nonn

AUTHOR

Ilya Gutkovskiy, Mar 27 2017

STATUS

approved

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Last modified October 16 16:08 EDT 2019. Contains 328101 sequences. (Running on oeis4.)