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

0,7

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

Reinhard Zumkeller, Rows n = 0..125 of triangle, flattened

Index entries for triangles and arrays related to Pascal's triangle

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

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

Cf. A105321, A249304, A249307, A249133.

Sequence in context: A254631 A029385 A185155 * A026536 A046213 A215625

Adjacent sequences: A249092 A249093 A249094 * A249096 A249097 A249098

Keyword

nonn,tabf

Author

Reinhard Zumkeller, Nov 14 2014

Status

approved