OFFSET
0,3
LINKS
G. C. Greubel, Table of n, a(n) for the first 25 rows
FORMULA
T(0,0) = 1;
T(n,0) = 1;
T(n,k) = T(n-1, k-1) + T(n-1, k) if k < n;
T(n,n) = (Sum_{j=0..n-1} Sum_{i=0..j} T(j,i)) + Sum_{i=0..n-1} T(n,i) [i.e., sum of all earlier terms of the triangle].
T(n,n) = (4^n)/2 for n > 0;
T(n,n) = 2*Sum_{i=0..n-1} T(n,i).
EXAMPLE
T(2,1) = T(1,0) + T(1,1) = 1 + 2 = 3;
T(2,2) = T(0,0) + T(1,0) + T(1,1) + T(2,0) + T(2,1) = 1 + 1 + 2 + 1 + 3 = 8.
From G. C. Greubel, Oct 09 2016: (Start)
The triangle is:
1;
1, 2;
1, 3, 8;
1, 4, 11, 32;
1, 5, 15, 43, 128;
1, 6, 20, 58, 171, 512;
... (End)
MATHEMATICA
T[0, 0] := 1; T[n_, 0] := 1; T[n_, k_] := T[n - 1, k] + T[n - 1, k - 1]; T[n_, n_] := 2^(2*n - 1); Table[T[n, k], {n, 0, 5}, {k, 0, n}] (* G. C. Greubel, Oct 09 2016 *)
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Jose Ramon Real, Dec 04 2007
STATUS
approved