OFFSET
0,7
COMMENTS
Length of row n = 2*n+1;
T(n,2*k) = A007318(n,k), 0 <= k <= n;
T(n,2*k+1) = A007318(n-1,k-1), n > 0 and 0 <= k < n;
T(n,k) = T(n-1,k-2) + T(n-1,k), n > 0 and 2 <= k <= n-2;
T(n,2*k) = T(n-1,2*k) + T(n-1,2*(k-1)), k = 0..n;
T(n,2*k+1) = T(n-2,2*k), k = 0..n-1;
T(n,n) = A128014(n);
A105321(n) = number of odd terms in row n;
A249304(n) = number of even terms in row n;
T(n,k) mod 2 = A249133(n,k).
LINKS
FORMULA
T(n,2*k) = T(n,2*k-1) + T(n,2*k+1), 0 < k < n.
EXAMPLE
The triangle begins:
. 0: 1
. 1: 1 1 1
. 2: 1 1 2 1 1
. 3: 1 1 3 2 3 1 1
. 4: 1 1 4 3 6 3 4 1 1
. 5: 1 1 5 4 10 6 10 4 5 1 1
. 6: 1 1 6 5 15 10 20 10 15 5 6 1 1
. 7: 1 1 7 6 21 15 35 20 35 15 21 6 7 1 1
. 8: 1 1 8 7 28 21 56 35 70 35 56 21 28 7 8 1 1
. 9: 1 1 9 8 36 28 84 56 126 70 126 56 84 28 36 8 9 1 1 .
MATHEMATICA
t[n_, k_] := If[n > 1 && 1 < k < 2*n - 1, If[EvenQ[k], t[n - 1, k] + t[n - 1, k - 2], t[n - 1, k - 1]], 1]; Grid[Table[t[n, k], {n, 0, 9}, {k, 0, 2*n}]] (* L. Edson Jeffery, Nov 30 2014 *)
PROG
(Haskell)
import Data.List (transpose)
a249095 n k = a249095_tabf !! n !! k
a249095_row n = a249095_tabf !! n
a249095_tabf = [1] : map (concat . transpose)
(zipWith ((. return) . (:)) (tail a007318_tabl) a007318_tabl)
CROSSREFS
KEYWORD
nonn,tabf
AUTHOR
Reinhard Zumkeller, Nov 14 2014
STATUS
approved