A249095 Triangle read by rows: interleaving successive pairs of rows of Pascal's triangle.

1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 3, 2, 3, 1, 1, 1, 1, 4, 3, 6, 3, 4, 1, 1, 1, 1, 5, 4, 10, 6, 10, 4, 5, 1, 1, 1, 1, 6, 5, 15, 10, 20, 10, 15, 5, 6, 1, 1, 1, 1, 7, 6, 21, 15, 35, 20, 35, 15, 21, 6, 7, 1, 1, 1, 1, 8, 7, 28, 21, 56, 35, 70, 35, 56, 21, 28, 7, 8, 1, 1

Offset: 0

Author

Reinhard Zumkeller, Nov 14 2014

Keywords

• nonn
• tabf

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).

Examples

			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 .
		

Links

• Reinhard Zumkeller, Rows n = 0..125 of triangle, flattened
• Index entries for triangles and arrays related to Pascal's triangle

Crossrefs

Cf. A005408 (row lengths), A128014 (central terms), A003945 (row sums), A249111 (partial sums per row), A007318 (Pascal).

Cf. A105321, A249304, A249307, A249133.

Programs

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)

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 *)

Formula

T(n,2*k) = T(n,2*k-1) + T(n,2*k+1), 0 < k < n.