OFFSET
0,4
COMMENTS
Row sums = A081659: (1, 2 4, 6, 9, 13, 19, 28, ...).
LINKS
G. C. Greubel, Rows n = 0..100 of triangle, flattened
FORMULA
T(n,k) = 1 + abs( ((1+(-1)^(n-k))/2)*binomial((n+k)/2, (n-k)/2)*cos((n-k)*Pi/2) ), with T(n,n) = 1. - G. C. Greubel, Nov 20 2019
EXAMPLE
First few rows of the triangle:
1;
1, 1;
2, 1, 1;
1, 3, 1, 1;
2, 1, 4, 1, 1;
1, 4, 1, 5, 1, 1;
2, 1, 7, 1, 6, 1, 1;
1, 5, 1, 11, 1, 7, 1, 1;
2, 1, 11, 1, 16, 1, 8, 1, 1;
...
MAPLE
T:= proc(n, k) option remember;
if k=n then 1
else 1 + abs( ((1+(-1)^(n-k))/2)*binomial((n+k)/2, (n-k)/2)*cos((n-k)*Pi/2) )
fi; end:
seq(seq(T(n, k), k=0..n), n=0..15); # G. C. Greubel, Nov 20 2019
MATHEMATICA
T[n_, k_]:= T[n, k]= If[k==n, 1, 1 + Abs[Simplify[((1+(-1)^(n-k))/2)* Binomial[(n+k)/2, (n-k)/2]*Cos[(n-k)*Pi/2]]] ]; Table[T[n, k], {n, 0, 15}, {k, 0, n}]//Flatten (* G. C. Greubel, Nov 20 2019 *)
PROG
(Sage)
@CachedFunction
def T(n, k):
if (k==n): return 1
else: return 1 + abs( ((1+(-1)^(n-k))/2)*binomial((n+k)/2, (n-k)/2)*cos((n-k)*pi/2) )
[[T(n, k) for k in (0..n)] for n in (0..15)] # G. C. Greubel, Nov 20 2019
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Gary W. Adamson, Nov 23 2007
EXTENSIONS
More terms added and offset changed by G. C. Greubel, Nov 20 2019
STATUS
approved