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%I #16 Jul 19 2024 14:36:13
%S 1,1,1,1,7,13,45,91,233,477,1079,2205,4709,10299,22393,52005,125055,
%T 310373,799677,2096699,5556681,14806685,39417431,104570549,276027337,
%U 724183555,1887993925,4891368373,12595644523,32252683453,82146468813,208225916203,525472131209
%N Number of n-color complete compositions of n.
%C These are integer compositions whose set of parts covers an initial interval and contains k colors of each part k.
%e a(6) = 13 counts: (1,1,1,1,1,1) and the 12 permutations of parts 1, 1, 2_a, and 2_b.
%o (PARI)
%o colr(x,y)={my(r=y-x+1, v=[x..y], z = vector(r*(r+(1+(x-1)*2))/2), k=1); for(i=1,#v,for(j=1,v[i],z[k]=v[i]; k++)); return(z)}
%o C_x(s,N)={my(x='x+O('x^N), g=if(#s <1,1, sum(i=1,#s, C_x(s[^i],N) * x^(s[i]) )/(1-sum(i=1,#s, x^(s[i]))))); return(g)}
%o B_x(N)={my(x='x+O('x^N), j=1, h=0, s=colr(1,j)); while(vecsum(s) <= N, h += C_x(s,N+1); j++;s=colr(1,j)); my(a = Vec(h)); vector(N, i, a[i])}
%o B_x(25)
%Y Cf. A003242, A011782, A107428, A107429, A374147, A374726, A374728.
%K nonn
%O 1,5
%A _John Tyler Rascoe_, Jul 17 2024
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