OFFSET
0,9
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.
LINKS
Alois P. Heinz, Rows n = 0..200, flattened
FORMULA
G.f.: A(x,y) = Sum_{i>=0} i! * x^(i*(i+1)/2) * y^i / Product_{j=1..i} (1 - y * Sum_{k=1..j} (k*x^k)).
EXAMPLE
Triangle begins:
k=0 1 2 3 4 5 6 7 8 9
n=0 [1]
n=1 [0, 1]
n=2 [0, 0, 1]
n=3 [0, 0, 2, 1]
n=4 [0, 0, 0, 4, 1]
n=5 [0, 0, 0, 4, 6, 1]
n=6 [0, 0, 0, 6, 12, 8, 1]
n=7 [0, 0, 0, 0, 26, 24, 10, 1]
n=8 [0, 0, 0, 0, 24, 68, 40, 12, 1]
n=9 [0, 0, 0, 0, 18, 112, 140, 60, 14, 1]
...
T(5,3) = 4: (1,2_a,2_a), (1,2_a,2_b), (1,2_b,2_a), (1,2_b,2_b).
T(5,4) = 6: (1,2_a,1,1), (1,1,2_a,1), (1,1,1,2_a), (1,2_b,1,1), (1,1,2_b,1), (1,1,1,2_b).
T(5,5) = 1: (1,1,1,1,1).
MAPLE
b:= proc(n, i) option remember; expand(`if`(n=0, 1,
add(b(n-j, max(i, j))*j*x, j=1..min(i+1, n))))
end:
T:= (n, k)-> coeff(b(n, 0), x, k):
seq(seq(T(n, k), k=0..n), n=0..11); # Alois P. Heinz, May 03 2026
PROG
(PARI) A_xy(N) = {my(x='x+O('x^(N+1)), A = sum(i=0, N, i! * x^(i*(i+1)/2) * y^i / prod(j=1, i, 1 - y*sum(k=1, j, k*x^k)))); vector(N, n, Vecrev(polcoeff(A, n-1)))}
CROSSREFS
KEYWORD
AUTHOR
John Tyler Rascoe, May 02 2026
STATUS
approved