A332031 G.f.: Sum_{k>=1} k! * x^(k^2) / (1 - x^k).

1, 1, 1, 3, 1, 3, 1, 3, 7, 3, 1, 9, 1, 3, 7, 27, 1, 9, 1, 27, 7, 3, 1, 33, 121, 3, 7, 27, 1, 129, 1, 27, 7, 3, 121, 753, 1, 3, 7, 147, 1, 729, 1, 27, 127, 3, 1, 753, 5041, 123, 7, 27, 1, 729, 121, 5067, 7, 3, 1, 873, 1, 3, 5047, 40347, 121, 729, 1, 27, 7, 5163, 1, 41073, 1, 3, 127

Offset: 1

Author

Ilya Gutkovskiy, Feb 05 2020

Keywords

• nonn
• look

Comments

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

Number of compositions of n that are either singular compositions (just [n]), or where the difference between successive parts is always 2. - Antti Karttunen, Dec 15 2021

Examples

			a(12) = 9 because we have [12], [7, 5], [6, 4, 2], [6, 2, 4], [5, 7], [4, 6, 2], [4, 2, 6], [2, 6, 4] and [2, 4, 6].
		

Links

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

Crossrefs

Cf. A000142, A008578 (positions of 1's), A038548, A066839, A107461.

Coincides with A332032 on odd numbers.

Programs

Mathematica

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

PARI

A332031(n) = sumdiv(n, d, (d<=(n/d)) * d!); \\ Antti Karttunen, Dec 15 2021

Formula

From Antti Karttunen, Dec 15 2021: (Start)

a(n) = Sum_{d|n, d <= n/d} d!.

a(2n-1) = A332032(2n-1) for all n >= 1.

(End)