OFFSET
0,3
FORMULA
EXAMPLE
Triangle starts:
0 : [1]
1 : [0, 4]
2 : [0, 10, 10]
3 : [0, 20, 40, 20]
4 : [0, 35, 135, 100, 35]
5 : [0, 56, 340, 420, 200, 56]
6 : [0, 84, 784, 1370, 950, 350, 84]
7 : [0, 120, 1596, 3900, 3580, 1800, 560, 120]
8 : [0, 165, 3070, 9905, 11835, 7425, 3045, 840, 165]
9 : [0, 220, 5500, 23180, 34780, 27020, 13360, 4760, 1200, 220]
10 : [0, 286, 9466, 50480, 94030, 87992, 51886, 21840, 7020, 1650, 286]
...
MAPLE
b:= proc(n, i) option remember; expand(`if`(n=0, 1, `if`(i<1, 0, add(
b(n-i*j, min(n-i*j, i-1))*binomial(i*(i^2+6*i+11)/6+j, j)*x^j, j=0..n/i))))
end:
T:= (n, k)-> coeff(b(n$2), x, k):
seq(seq(T(n, k), k=0..n), n=0..10); # Alois P. Heinz, Mar 22 2025
MATHEMATICA
b[n_, i_] := b[n, i] = Expand[If[n == 0, 1, If[i < 1, 0, Sum[b[n - i*j, Min[n - i*j, i - 1]]*Binomial[i*(i^2 + 6*i + 11)/6 + j, j]*x^j, {j, 0, n/i}]]]];
T[n_, k_] := Coefficient[b[n, n], x, k];
Table[Table[T[n, k], {k, 0, n}], {n, 0, 10}] // Flatten (* Jean-François Alcover, Apr 17 2025, after Alois P. Heinz *)
PROG
(Python)
from sympy import binomial
from sympy.utilities.iterables import partitions
colors = 4 - 1 # the number of colors - 1
def t_row( n):
if n == 0 : return [1]
t = list( [0] * n)
for p in partitions( n):
fact = 1
s = 0
for k in p :
s += p[k]
fact *= binomial( binomial( k + colors, colors) + p[k] - 1, p[k])
if s > 0 :
t[s - 1] += fact
return [0] + t
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Peter Dolland, Mar 22 2025
STATUS
approved
