cp's OEIS Frontend

A374147 Number of complete Carlitz compositions of n.

%I A374147 #12 Jun 30 2024 09:10:00
%S A374147 1,0,2,1,1,8,7,9,20,49,72,115,202,349,695,1171,2009,3530,6203,10818,
%T A374147 19320,33961,59449,104349,183370,321635,564081,992513,1741441,3057547,
%U A374147 5363570,9410785,16516575,28967505,50798456,89106542,156276871,274037619,480437247,842350671,1476760717,2588651452,4537418431,7952741429,13938276465
%N A374147 Number of complete Carlitz compositions of n.
%C A374147 These are integer compositions such that no adjacent parts are equal and their set of parts covers an initial interval.
%F A374147 G.f.: Sum_{k>0} Ca({1..k},x) where Ca({s},x) = Sum_{i in {s}} ( (Ca({s}-{i},x)*x^i)/(1 + x^i) )/(1 - Sum_{i in {s}} ( (x^i)/(1 + x^i) )) is the g.f. for Carlitz compositions such that their set of parts equals {s} with Ca({},x) = 1.
%e A374147 a(7) = 7 counts: (1,2,1,3), (1,2,3,1), (1,3,2,1), (1,3,1,2), (2,1,3,1), (3,2,1,2), (1,2,1,2,1).
%o A374147 (PARI)
%o A374147 Ca_x(s,N)={my(x='x+O('x^N), g=if(#s <1,1, sum(i=1,#s, (Ca_x(s[^i],N) * x^(s[i])/(1+x^(s[i]))))/(1-sum(i=1,#s, (x^(s[i]))/(1+x^(s[i])))))); return(g)}
%o A374147 B_x(N)={my(x='x+O('x^N), j=1, h=0); while((j*(j+1))/2 <= N, h += Ca_x([1..j],N+1); j+=1); my(a = Vec(h)); vector(N,i,a[i])}
%o A374147 B_x(45)
%Y A374147 Cf. A003242, A107428, A107429.
%K A374147 nonn
%O A374147 1,3
%A A374147 _John Tyler Rascoe_, Jun 28 2024