A332032 G.f.: Sum_{k>=1} k! * x^(k^2) / (1 - x^(2*k)).

1, 0, 1, 2, 1, 0, 1, 2, 7, 0, 1, 2, 1, 0, 7, 26, 1, 0, 1, 2, 7, 0, 1, 26, 121, 0, 7, 2, 1, 0, 1, 26, 7, 0, 121, 722, 1, 0, 7, 26, 1, 0, 1, 2, 127, 0, 1, 746, 5041, 0, 7, 2, 1, 0, 121, 26, 7, 0, 1, 722, 1, 0, 5047, 40346, 121, 0, 1, 2, 7, 0, 1, 746, 1, 0, 127

Offset: 1

Author

Ilya Gutkovskiy, Feb 05 2020

Keywords

• nonn

Comments

Number of compositions (ordered partitions) of n into distinct odd parts, where every odd part between the largest and smallest appears.

Examples

			a(9) = 7 because we have [9], [5, 3, 1], [5, 1, 3], [3, 5, 1], [3, 1, 5], [1, 5, 3] and [1, 3, 5].
a(12) = 2 because we have [7, 5], and [5, 7]. - _Antti Karttunen_, Dec 15 2021
		

Links

• Antti Karttunen, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A016825 (positions of 0's), A032021, A034178, A038548, A107461, A220400.

Coincides with A332031 on odd numbers.

Programs

Mathematica

nmax = 75; CoefficientList[Series[Sum[k! x^(k^2)/(1 - x^(2 k)), {k, 1, nmax}], {x, 0, nmax}], x] // Rest

PARI

A332032(n) = sumdiv(n, d, if(!((n+d)%2) && !((n+(n/d))%2) && (d<=(n/d)), d!, 0)); \\ Antti Karttunen, Dec 15 2021

Formula

From Antti Karttunen, Dec 15 2021: (Start)

a(n) = Sum_{d|n, d <= n/d, and both d and n/d are of the same parity as n} d!.

a(2n-1) = A332031(2n-1) for all n >= 1.

(End)