OFFSET
0,45
COMMENTS
T(n,k) is defined for all n,k >= 0. The triangle contains in each row n only the terms for k=0 and then up to the last positive T(n,k) (if it exists).
LINKS
Alois P. Heinz, Rows n = 0..500, flattened
EXAMPLE
Triangle T(n,k) begins:
1 ;
0 ;
0 ;
0 ;
0, 1 ;
0 ;
0, 1 ;
0 ;
0, 0, 1 ;
0, 1 ;
0, 1, 1 ;
0 ;
0, 0, 1, 1 ;
0, 0, 1 ;
0, 1, 1, 1 ;
0, 1, 1 ;
0, 0, 1, 1, 1 ;
0, 0, 0, 1 ;
0, 0, 2, 2, 1 ;
0, 0, 2, 1 ;
0, 0, 2, 1, 1, 1 ;
...
MAPLE
h:= proc(n) option remember; `if`(n=0, 0,
`if`(numtheory[bigomega](n)=2, n, h(n-1)))
end:
b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
`if`(i>n, 0, expand(x*b(n-i, h(min(n-i, i)))))+b(n, h(i-1))))
end:
T:= n-> (p-> seq(coeff(p, x, i), i=0..max(0, degree(p))))(b(n, h(n))):
seq(T(n), n=0..32);
MATHEMATICA
h[n_] := h[n] = If[n == 0, 0,
If[PrimeOmega[n] == 2, n, h[n-1]]];
b[n_, i_] := b[n, i] = If[n == 0, 1, If[i < 1, 0,
If[i > n, 0, Expand[x*b[n-i, h[Min[n-i, i]]]]] + b[n, h[i-1]]]];
T[n_] := Table[Coefficient[#, x, i], {i, 0, Max[0, Exponent[#, x]]}]&[b[n, h[n]]];
Table[T[n], {n, 0, 32}] // Flatten (* Jean-François Alcover, Aug 19 2021, after Alois P. Heinz *)
CROSSREFS
KEYWORD
nonn,tabf
AUTHOR
Alois P. Heinz, May 19 2021
STATUS
approved