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proposed

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 *)

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proposed

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 *)

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A249304(n) = number of even terms in row n.;

T(n,k) mod 2 = A249133(n,k).

Cf. A105321, A249304, A249307, A249133.

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.

Reinhard Zumkeller, <a href="/A249095/b249095.txt">Table of n, a(n) for Rows n = 0..10200125 of triangle, flattened</a>

<a href="/index/Pas#Pascal">Index entries for triangles and arrays related to Pascal's triangle</a>

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

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 .

(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)

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

Cf. A105321, A249304, A249307.

nonn,changed,tabf

allocated for Reinhard ZumkellerTriangle 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

0,7

Reinhard Zumkeller, <a href="/A249095/b249095.txt">Table of n, a(n) for n = 0..10200</a>

allocated

nonn

Reinhard Zumkeller, Nov 14 2014

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