A395480 Number of n-color compositions of n with parts in standard order.

1, 1, 1, 3, 5, 11, 27, 61, 145, 345, 839, 2033, 4983, 12259, 30313, 75297, 187649, 469211, 1176525, 2957347, 7450003, 18804669, 47548049, 120417569, 305401503, 775573657, 1971961879, 5019461283, 12789790601, 32620177385, 83271844905, 212752366315, 543992364541

Offset

0,4

Comments

A composition with parts in standard order satisfies the condition that for any part p > 1, the part p - 1 has already appeared and each part p in an n-color composition can be p different colors. All terms are odd.

Links

Alois P. Heinz, Table of n, a(n) for n = 0..2399

Formula

G.f.: Sum_{i>=0} i! * x^(i*(i+1)/2) / Product_{j=1..i} (1 - Sum_{k=1..j} (k*x^k)).

Example

a(4) = 5: (1,1,1,1), (1,2_a,1), (1,2_b,1), (1,1,2_a), (1,1,2_b).

Maple

b:= proc(n, i) option remember; `if`(n=0, 1,
      add(b(n-j, max(i, j))*j, j=1..min(i+1, n)))
    end:
a:= n-> b(n, 0):
seq(a(n), n=0..32);  # Alois P. Heinz, Apr 26 2026

Prog

(PARI) A_x(N) = {my(x='x+O('x^(N+1)), A = sum(i=0, N, i! * x^(i*(i+1)/2) / prod(j=1, i, 1 - sum(k=1, j, k*x^k)))); Vec(A)}

Crossrefs

Row sums of A394777.

Cf. A000110, A383253, A383713.

Sequence in context: A288828 A088357 A369344 * A204857 A292855 A374572

Adjacent sequences: A395477 A395478 A395479 * A395481 A395482 A395483

Keyword

nonn,easy

Author

John Tyler Rascoe, Apr 25 2026

Status

approved