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

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, 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, 2, 2, 493, 1, 44, 26, 2, 2, 50, 26, 2, 5, 2, 2, 698, 2, 50, 30, 2, 2

Offset

0,4

Links

Alois P. Heinz, Table of n, a(n) for n = 0..20000 (first 3001 terms from Michael De Vlieger)

Formula

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

a(2^k) = 1.

Maple

with(numtheory):
a:= proc(n) option remember; local b, l; l, b:= sort(
      [select(x-> is((n/x):: odd), divisors(n))[]]),
      proc(m, i) option remember; `if`(m=0, 1, `if`(i<1, 0,
        b(m, i-1)+`if`(l[i]>m, 0, b(m-l[i], i))))
      end; b(n, nops(l))
    end:
seq(a(n), n=0..100);  # Alois P. Heinz, Dec 06 2021

Mathematica

a[n_] := SeriesCoefficient[Product[1/(1 - Boole[OddQ[n/d]] x^d), {d, Divisors[n]}], {x, 0, n}]; Table[a[n], {n, 0, 80}]

Prog

(Magma) [1] cat [#RestrictedPartitions(n, {d:d in Divisors(n)|IsOdd(n div d)}):n in [1..80]]; // Marius A. Burtea, Sep 20 2019

Crossrefs

Cf. A018818, A038550 (positions of 2's), A171565.

Sequence in context: A098965 A290087 A016443 * A300648 A318450 A120256

Adjacent sequences: A327637 A327638 A327639 * A327641 A327642 A327643

Keyword

nonn

Author

Ilya Gutkovskiy, Sep 20 2019

Status

approved