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

1,4

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

Links

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

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)

Example

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].

Mathematica

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

Prog

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

Crossrefs

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

Coincides with A332032 on odd numbers.

Sequence in context: A176246 A046933 A185091 * A023511 A035628 A187562

Adjacent sequences: A332028 A332029 A332030 * A332032 A332033 A332034

Keyword

nonn,look

Author

Ilya Gutkovskiy, Feb 05 2020

Status

approved