OFFSET
0,3
FORMULA
T(n,n) = binomial(n + 2, 2) = A000217(n + 1).
T(n,1) = 3 for n >= 1.
Sum_{k=0..n} (-1)^k * T(n,k) = A022598(n). - Alois P. Heinz, Mar 27 2025
EXAMPLE
Triangle starts:
0 : [1]
1 : [0, 3]
2 : [0, 3, 6]
3 : [0, 3, 9, 10]
4 : [0, 3, 15, 18, 15]
5 : [0, 3, 18, 36, 30, 21]
6 : [0, 3, 24, 55, 66, 45, 28]
7 : [0, 3, 27, 81, 114, 105, 63, 36]
8 : [0, 3, 33, 108, 189, 195, 153, 84, 45]
9 : [0, 3, 36, 145, 276, 348, 298, 210, 108, 55]
10 : [0, 3, 42, 180, 405, 552, 558, 423, 276, 135, 66]
...
MAPLE
b:= proc(n, i) option remember; expand(`if`(n=0, 1, `if`(i<1, 0,
add(x^j*b(n-i*j, min(n-i*j, i-1))*(j+2)*(j+1)/2, 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 27 2025
MATHEMATICA
b[n_, i_] := b[n, i] = Expand[If[n == 0, 1, If[i < 1, 0, Sum[x^j*b[n-i*j, Min[n-i*j, i-1]]*(j+2)*(j+1)/2, {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, Jul 30 2025, after Alois P. Heinz *)
PROG
(Python)
from sympy import binomial
from sympy.utilities.iterables import partitions
kinds = 3 - 1 # the number of part kinds - 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( kinds + p[k], kinds)
if s > 0 :
t[s - 1] += fact
return [0] + t
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Peter Dolland, Mar 27 2025
STATUS
approved
