OFFSET
0,2
COMMENTS
Row sums = A048487: (1, 6, 16, 36, 76, 156, ...).
LINKS
FORMULA
Triangle, given borders = (1, 3, 5, 7, 9, ...); apply Pascal's rule T(n,k) = T(n-1,k) P T(n-1,k-1).
T(n,k) = A051601(n,k) + A051597(n,k); T(n,k) mod 2 = A047999(n,k). - Reinhard Zumkeller, Nov 23 2012
Closed-form formula for arbitrary left and right borders of Pascal like triangle see A228196. - Boris Putievskiy, Aug 19 2013
EXAMPLE
First few rows of the triangle:
1;
3, 3;
5, 6, 5;
7, 11, 11, 7;
9, 18, 22, 18, 9;
11, 27, 40, 40, 27, 11;
13, 38, 67, 80, 67, 38, 13;
...
MAPLE
T:= proc(n, k) option remember;
`if`(k<0 or k>n, 0,
`if`(k=0 or k=n, 2*n+1,
T(n-1, k-1) + T(n-1, k) ))
end:
seq(seq(T(n, k), k=0..n), n=0..14); # Alois P. Heinz, May 26 2013
MATHEMATICA
NestList[Append[Prepend[Map[Apply[Plus, #] &, Partition[#, 2, 1]], #[[1]] + 2], #[[1]] + 2] &, {1}, 10] // Grid (* Geoffrey Critzer, May 26 2013 *)
T[n_, k_] := Binomial[n, k-1] + Binomial[n, k] + 2 Binomial[n, k+1] + Binomial[n, n-k+1];
Table[T[n, k], {n, 0, 14}, {k, 0, n}] // Flatten (* Jean-François Alcover, Mar 07 2021 *)
PROG
(Haskell)
a134636 n k = a134636_tabl !! n !! k
a134636_row n = a134636_tabl !! n
a134636_tabl = iterate (\row -> zipWith (+) ([2] ++ row) (row ++ [2])) [1]
-- Reinhard Zumkeller, Nov 23 2012
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Gary W. Adamson, Nov 04 2007
EXTENSIONS
Offset changed by Reinhard Zumkeller, Nov 23 2012
STATUS
approved