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A332031
G.f.: Sum_{k>=1} k! * x^(k^2) / (1 - x^k).
2
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
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
KEYWORD
nonn,look
AUTHOR
Ilya Gutkovskiy, Feb 05 2020
STATUS
approved