OFFSET
0,3
LINKS
Alois P. Heinz, Rows n = 0..200, flattened
FORMULA
T(n,n) = n + 1.
T(n,1) = 2 for n >= 1.
Sum_{k=0..n} (-1)^k * T(n,k) = A022597(n). - Alois P. Heinz, Mar 27 2025
EXAMPLE
Triangle starts:
0 : [1]
1 : [0, 2]
2 : [0, 2, 3]
3 : [0, 2, 4, 4]
4 : [0, 2, 7, 6, 5]
5 : [0, 2, 8, 12, 8, 6]
6 : [0, 2, 11, 18, 17, 10, 7]
7 : [0, 2, 12, 26, 28, 22, 12, 8]
8 : [0, 2, 15, 34, 46, 38, 27, 14, 9]
9 : [0, 2, 16, 46, 64, 66, 48, 32, 16, 10]
10 : [0, 2, 19, 56, 94, 100, 86, 58, 37, 18, 11]
...
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+1), j=0..n/i))))
end:
T:= (n, k)-> coeff(b(n$2), x, k):
seq(seq(T(n, k), k=0..n), n=0..11); # 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 + 1), {j, 0, n/i}]]]];
T[n_, k_] := Coefficient[b[n, n], x, k];
Table[Table[T[n, k], {k, 0, n}], {n, 0, 11}] // Flatten (* Jean-François Alcover, Apr 19 2025, after Alois P. Heinz *)
PROG
(Python)
from sympy.utilities.iterables import partitions
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 *= 1 + p[k]
if s > 0 :
t[s - 1] += fact
return [0] + t
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Peter Dolland, Mar 27 2025
STATUS
approved
