OFFSET
0,7
LINKS
G. C. Greubel, Rows n = 0..50, flattened
FORMULA
From G. C. Greubel, Oct 23 2023: (Start)
Sum_{k=0..n} T(n, k) = A000079(n).
Sum_{k=0..2*n-1} T(n, k) = A000918(n+1), n >= 1.
Sum_{k=0..2*n} (-1)^k*T(n, k) = 1. (End)
EXAMPLE
First few rows of the triangle are;
1;
1, 1, 1;
1, 1, 2, 2, 1;
1, 1, 3, 3, 3, 3, 1;
1, 1, 4, 4, 6, 6, 4, 4, 1;
1, 1, 5, 5, 10, 10, 10, 10, 5, 5, 1;
1, 1, 6, 6, 15, 15, 20, 20, 15, 15, 6, 6, 1;
1, 1, 7, 7, 21, 21, 35, 35, 35, 35, 21, 21, 7, 7, 1;
...
MATHEMATICA
row[n_]:= Append[Table[Binomial[n, k], {k, 0, n-1}, {2}], 1]//Flatten;
Table[row[n], {n, 0, 7}]//Flatten (* Jean-François Alcover, Aug 02 2019 *)
PROG
(Sage)
@CachedFunction
def T(n, k): # Triangle in centered form.
if abs(k) > n: return 0
if n == k: return 1
even = lambda n: 1 if 2.divides(n) else 0
odd = lambda n: 1 if 2.divides(n+1) else 0
return T(n-1, k-1) + odd(n-k)*T(n-1, k) + even(n-k)*T(n-1, k+1)
for n in (0..7): [T(n, k) for k in (-n..n)] # Peter Luschny, Nov 22 2013
(Magma)
A140751:=func< n, k | k mod 2 eq 0 select Binomial(n, Floor(k/2)) else k mod 2 eq 1 select Binomial(n, Floor((k-1)/2)) else 0 >;
[A140751(n, k): k in [0..2*n], n in [0..12]]; // G. C. Greubel, Oct 23 2023
CROSSREFS
KEYWORD
nonn,tabf
AUTHOR
Gary W. Adamson & Roger L. Bagula, May 26 2008
STATUS
approved