A374727 Number of n-color complete compositions of n.

1, 1, 1, 1, 7, 13, 45, 91, 233, 477, 1079, 2205, 4709, 10299, 22393, 52005, 125055, 310373, 799677, 2096699, 5556681, 14806685, 39417431, 104570549, 276027337, 724183555, 1887993925, 4891368373, 12595644523, 32252683453, 82146468813, 208225916203, 525472131209

Offset: 1

Author

John Tyler Rascoe, Jul 17 2024

Keywords

• nonn

Comments

These are integer compositions whose set of parts covers an initial interval and contains k colors of each part k.

Examples

			a(6) = 13 counts: (1,1,1,1,1,1) and the 12 permutations of parts 1, 1, 2_a, and 2_b.
		

Crossrefs

Cf. A003242, A011782, A107428, A107429, A374147, A374726, A374728.

Programs

PARI

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)}
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)}
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])}
B_x(25)