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

0,3

Links

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

Formula

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

Example

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

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

Prog

(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)

Crossrefs

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

Sequence in context: A124329 A144520 A101399 * A320650 A018109 A341020

Adjacent sequences: A383172 A383173 A383174 * A383176 A383177 A383178

Keyword

nonn,easy

Author

John Tyler Rascoe, Apr 18 2025

Status

approved