A330773 Number of perfect compositions of n.
1, 1, 1, 3, 1, 5, 1, 11, 3, 5, 1, 27, 1, 5, 5, 49, 1, 27, 1, 27, 5, 5, 1, 163, 3, 5, 11, 27, 1, 49, 1, 261, 5, 5, 5, 231, 1, 5, 5, 163, 1, 49, 1, 27, 27, 5, 1, 1109, 3, 27, 5, 27, 1, 163, 5, 163, 5, 5, 1, 435, 1, 5, 27, 1631, 5, 49, 1, 27, 5, 49, 1, 2055, 1, 5, 27, 27, 5, 49, 1
Offset: 0
Examples
a(7) = 11 because the perfect compositions are 1111111, 1222, 2221, 1114, 4111, 124, 142, 214, 241, 412, 421. For example, 241 generates the compositions of 1,...,6: 1,2,21,4,41,24.
Links
- Alois P. Heinz, Table of n, a(n) for n = 0..20000
- A. O. Munagi, Perfect Compositions of Numbers, J. Integer Seq. 23 (2020), art. 20.5.1.
Programs
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Maple
b:= proc(n) option remember; expand(x*(1+add(b(n/d), d=numtheory[divisors](n) minus {1, n}))) end: a:= n-> (p-> add(coeff(p, x, i)*i!, i=1..degree(p)))(b(n+1)): seq(a(n), n=0..100); # Alois P. Heinz, Jan 15 2020 -
Mathematica
b[n_] := b[n] = x(1+Sum[b[n/d], {d, Divisors[n]~Complement~{1, n}}]); a[n_] := With[{p = b[n+1]}, Sum[Coefficient[p, x, i] i!, {i, Exponent[p, x]}]]; a /@ Range[0, 100] (* Jean-François Alcover, Nov 17 2020, after Alois P. Heinz *)
Formula
a(1)=1, a(n) = Sum_{k=1..Omega(n+1)} k! * A251683(n+1,k), n>1.
Comments