A383175 Number of compositions of n such that any fixed point k can be k different colors.

1, 1, 2, 5, 10, 22, 48, 101, 213, 450, 945, 1961, 4064, 8385, 17242, 35332, 72141, 146924, 298552, 605377, 1225277, 2475912, 4995754, 10067848, 20267680, 40762951, 81916919, 164504411, 330155437, 662265817, 1327860471, 2661376529, 5332341881, 10680912173

Offset: 0

Author

John Tyler Rascoe, Apr 18 2025

Keywords

• nonn
• easy

Examples

			a(3) = 5 counts: (3), (2,1), (1_a,2_a), (1_a,2_b), (1_a,1,1).
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..2000

Crossrefs

Cf. A011782, A088305, A238349, A238350, A238351, A335713, A352512.

Programs

Maple

b:= proc(n, i) option remember; `if`(n=0, 1, add(
     `if`(n<=i+j, ceil(2^(n-j-1)), b(n-j, i+1))*
     `if`(i=j, j, 1), j=1..n))
    end:
a:= n-> b(n, 1):
seq(a(n), n=0..33);  # Alois P. Heinz, Apr 18 2025

PARI

A_x(N) = {my(x='x+O('x^N)); Vec(1+sum(i=1,N, prod(j=1,i, j*x^j-x^j+x/(1-x))))}
A_x(30)

Formula

G.f.: 1 + Sum_{i>0} Product_{j=1..i} ( j*x^j - x^j + x/(1-x) ).