A395480 - OEIS

%I #19 May 12 2026 21:08:35

%S 1,1,1,3,5,11,27,61,145,345,839,2033,4983,12259,30313,75297,187649,

%T 469211,1176525,2957347,7450003,18804669,47548049,120417569,305401503,

%U 775573657,1971961879,5019461283,12789790601,32620177385,83271844905,212752366315,543992364541

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

%C 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.

%H Alois P. Heinz, <a href="/A395480/b395480.txt">Table of n, a(n) for n = 0..2399</a>

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

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

%p b:= proc(n, i) option remember; `if`(n=0, 1,

%p       add(b(n-j, max(i, j))*j, j=1..min(i+1, n)))

%p     end:

%p a:= n-> b(n,0):

%p seq(a(n), n=0..32);  # _Alois P. Heinz_, Apr 26 2026

%o (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)}

%Y Row sums of A394777.

%Y Cf. A000110, A383253, A383713.

%K nonn,easy

%O 0,4

%A _John Tyler Rascoe_, Apr 25 2026