cp's OEIS Frontend

A330773 Number of perfect compositions of n.

%I A330773 #24 Nov 17 2020 05:40:27
%S A330773 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,
%T A330773 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,
%U A330773 163,5,5,1,435,1,5,27,1631,5,49,1,27,5,49,1,2055,1,5,27,27,5,49,1
%N A330773 Number of perfect compositions of n.
%C A330773 A perfect composition of n is one whose sequence of parts contains one composition of every positive integer less than n.
%H A330773 Alois P. Heinz, <a href="/A330773/b330773.txt">Table of n, a(n) for n = 0..20000</a>
%H A330773 A. O. Munagi, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL23/Munagi/munagi13.html">Perfect Compositions of Numbers</a>, J. Integer Seq. 23 (2020), art. 20.5.1.
%F A330773 a(1)=1, a(n) = Sum_{k=1..Omega(n+1)} k! * A251683(n+1,k), n>1.
%e A330773 a(7) = 11 because the perfect compositions are 1111111, 1222, 2221, 1114, 4111, 124, 142, 214, 241, 412, 421.
%e A330773 For example, 241 generates the compositions of 1,...,6: 1,2,21,4,41,24.
%p A330773 b:= proc(n) option remember; expand(x*(1+add(b(n/d),
%p A330773        d=numtheory[divisors](n) minus {1, n})))
%p A330773     end:
%p A330773 a:= n-> (p-> add(coeff(p, x, i)*i!, i=1..degree(p)))(b(n+1)):
%p A330773 seq(a(n), n=0..100);  # _Alois P. Heinz_, Jan 15 2020
%t A330773 b[n_] := b[n] = x(1+Sum[b[n/d], {d, Divisors[n]~Complement~{1, n}}]);
%t A330773 a[n_] := With[{p = b[n+1]}, Sum[Coefficient[p, x, i] i!, {i, Exponent[p, x]}]];
%t A330773 a /@ Range[0, 100] (* _Jean-François Alcover_, Nov 17 2020, after _Alois P. Heinz_ *)
%Y A330773 Cf. A001222, A002033, A074206, A251683, A330774.
%K A330773 nonn,easy
%O A330773 0,4
%A A330773 _Augustine O. Munagi_, Dec 30 2019