OFFSET
0,5
LINKS
G. C. Greubel, Rows n = 0..100 of triangle, flattened
FORMULA
Sum_{k=0..n} T(n, k) = 2*(n-1)*(Sum_{k=0..n-1} T(n-1, k)) + 2 = A099760(n).
EXAMPLE
Triangle begins:
1;
1, 1;
1, 4, 1;
1, 12, 12, 1;
1, 30, 96, 30, 1;
1, 68, 564, 564, 68, 1;
MAPLE
T:=proc(n, k) if k=0 or n=k then 1 elif k>n then 0 else 2*(n-k)*T(n-1, k-1)+2*k*T(n-1, k) fi end: for n from 0 to 9 do [seq(T(n, k), k=0..n)] od; # gives the triangle row by row # Emeric Deutsch, Nov 16 2004
MATHEMATICA
T[n_, k_]:= T[n, k]= If[k==0 || k==n, 1, 2*(n-k)*T[n-1, k-1] +2*k*T[n-1, k]]; Table[T[n, k], {n, 0, 9}, {k, 0, n}]//Flatten (* G. C. Greubel, Sep 03 2019 *)
PROG
(PARI) T(n, k) = if(k==0 || k==n, 1, 2*(n-k)*T(n-1, k-1) + 2*k*T(n-1, k));
for(n=0, 9, for(k=0, n, print1(T(n, k), ", "))) \\ G. C. Greubel, Sep 03 2019
(Sage)
def T(n, k):
if (k==0 or k==n): return 1
else: return 2*k*T(n-1, k) + 2*(n-k)* T(n-1, k-1)
[[T(n, k) for k in (0..n)] for n in (0..9)] # G. C. Greubel, Sep 03 2019
(GAP)
T:= function(n, k)
if k=0 or k=n then return 1;
else return 2*(n-k)*T(n-1, k-1) + 2*k*T(n-1, k);
fi;
end;
Flat(List([0..9], n-> List([0..n], k-> T(n, k) ))); # G. C. Greubel, Sep 03 2019
CROSSREFS
KEYWORD
AUTHOR
Miklos Kristof, Nov 11 2004
EXTENSIONS
More terms from Emeric Deutsch, Nov 16 2004
STATUS
approved