OFFSET
1,5
COMMENTS
This triangle is dominated by the Eulerian numbers A008292.
LINKS
G. C. Greubel, Rows n = 1..50 of the triangle, flattened
FORMULA
Sum_{k=0..n} T(n, k) = (4^(n-1) + 2)/3 = A047849(n-1).
EXAMPLE
Triangle begins as:
1;
1, 1;
1, 4, 1;
1, 10, 10, 1;
1, 22, 40, 22, 1;
1, 46, 124, 124, 46, 1;
1, 94, 340, 496, 340, 94, 1;
1, 190, 868, 1672, 1672, 868, 190, 1;
1, 382, 2116, 5080, 6688, 5080, 2116, 382, 1;
1, 766, 4996, 14392, 23536, 23536, 14392, 4996, 766, 1;
MATHEMATICA
T[n_, k_]:= T[n, k]= If[k==1 || k==n, 1, 2*(T[n-1, k-1] +T[n-1, k])];
Table[T[n, k], {n, 10}, {k, n}]//Flatten (* modified by G. C. Greubel, Apr 13 2021 *)
a[0] = {1}; a[1] = {1, 1};
a[n_]:= a[n]= 2*Join[a[n-1], {-1/2}] + 2*Join[{-1/2}, a[n-1]];
Table[a[n], {n, 0, 10}]//Flatten (* Roger L. Bagula, Dec 09 2008 *)
PROG
(Magma)
function T(n, k)
if k eq 1 or k eq n then return 1;
else return 2*(T(n-1, k-1) + T(n-1, k));
end if; return T;
end function;
[T(n, k): k in [1..n], n in [1..12]]; // G. C. Greubel, Apr 13 2021
(Sage)
@CachedFunction
def T(n, k): return 1 if k==1 or k==n else 2*(T(n-1, k-1) + T(n-1, k))
flatten([[T(n, k) for k in (1..n)] for n in (1..12)]) # G. C. Greubel, Apr 13 2021
CROSSREFS
KEYWORD
AUTHOR
Roger L. Bagula, Sep 22 2008
EXTENSIONS
Edited by N. J. A. Sloane, Dec 11 2008
STATUS
approved