OFFSET
1,3
LINKS
FORMULA
EXAMPLE
Triangle T(n,k) begins:
: 1 ;
: 1, 2, 1 ;
: 1, 4, 6, 4, 1 ;
: 1, 6, 18, 24, 18, 6, 1 ;
: 1, 8, 36, 96, 120, 96, 36, 8, 1 ;
: 1, 10, 60, 240, 600, 720, 600, 240, 60, 10, 1 ;
: 1, 12, 90, 480, 1800, 4320, 5040, 4320, 1800, 480, 90, 12, 1 ;
MAPLE
b:= proc(s, c) option remember; (n-> `if`(n=0, c,
add(b(s minus {i}, c+x^(n-i)), i=s)))(nops(s))
end:
T:= n-> (p-> seq(coeff(p, x, i)/abs(i)!, i=1-n..n-1))(b({$1..n}, 0)):
seq(T(n), n=1..8);
# second Maple program:
egf:= k-> (t-> x^t/t!*hypergeom([2, t], [t+1], x))(abs(k)+1):
T:= (n, k)-> n! * coeff(series(egf(k), x, n+1), x, n):
seq(seq(T(n, k), k=1-n..n-1), n=1..8);
# third Maple program:
T:= (n, k)-> (t-> `if`(t<n, (n-t)*(n-1)!/t!, 0))(abs(k)):
seq(seq(T(n, k), k=1-n..n-1), n=1..8);
MATHEMATICA
T[n_, k_] := With[{t = Abs[k]}, If[t<n, (n-t)(n-1)!/t!, 0]];
Table[Table[T[n, k], {k, 1-n, n-1}], {n, 1, 8}] // Flatten (* Jean-François Alcover, Mar 25 2021, after 3rd Maple program *)
CROSSREFS
KEYWORD
nonn,tabf
AUTHOR
Alois P. Heinz, Feb 18 2019
STATUS
approved