A332031 - OEIS

%I #17 Dec 15 2021 17:46:52

%S 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,

%T 1,27,7,3,121,753,1,3,7,147,1,729,1,27,127,3,1,753,5041,123,7,27,1,

%U 729,121,5067,7,3,1,873,1,3,5047,40347,121,729,1,27,7,5163,1,41073,1,3,127

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

%C 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.

%C 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

%H Antti Karttunen, <a href="/A332031/b332031.txt">Table of n, a(n) for n = 1..10000</a>

%F From _Antti Karttunen_, Dec 15 2021: (Start)

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

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

%F (End)

%e 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].

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

%o (PARI) A332031(n) = sumdiv(n, d, (d<=(n/d)) * d!); \\ _Antti Karttunen_, Dec 15 2021

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

%Y Coincides with A332032 on odd numbers.

%K nonn,look

%O 1,4

%A _Ilya Gutkovskiy_, Feb 05 2020