A109446 Binomial coefficients C(n,k) with n-k even, read by rows.

1, 1, 1, 1, 3, 1, 1, 6, 1, 5, 10, 1, 1, 15, 15, 1, 7, 35, 21, 1, 1, 28, 70, 28, 1, 9, 84, 126, 36, 1, 1, 45, 210, 210, 45, 1, 11, 165, 462, 330, 55, 1, 1, 66, 495, 924, 495, 66, 1, 13, 286, 1287, 1716, 715, 78, 1, 1, 91, 1001, 3003, 3003, 1001, 91, 1, 15, 455, 3003, 6435, 5005

Offset: 0

Author

Philippe Deléham, Aug 27 2005

Keywords

• easy
• nonn
• tabf

Comments

Binomial(n,2(n-k-1)) is also the number of permutations avoiding both 123 and 132 with k descents, i.e., positions with w[i]>w[i+1]. - Lara Pudwell, Dec 19 2018

Examples

			Starred terms in Pascal's triangle (A007318), read by rows:
  1*;
  1,  1*;
  1*,  2,  1*;
  1,  3*,   3,  1*;
  1*,  4,  6*,   4,  1*;
  1,  5*,  10, 10*,   5,   1*;
  1*,  6, 15*,  20, 15*,    6,  1*;
  1,  7*,  21, 35*,  35,  21*,   7,  1*;
  1*,  8, 28*,  56, 70*,   56, 28*,   8, 1*;
  1,  9*,  36, 84*, 126, 126*,  84, 36*,  9, 1*;
Rows in A086645 (1; 1, 1; 1, 6, 1; ...) interspersed with rows in A103327 (1; 3, 1; 5, 10, 1; ...).
1; 1; 1, 1; 3, 1; 1, 6, 1; 5, 10, 1; 1, 15, 15, 1; 7, 35, 21, 1; ....
		

Links

• Alois P. Heinz, Rows n = 0..200, flattened
• M. Bukata, R. Kulwicki, N. Lewandowski, L. Pudwell, J. Roth, and T. Wheeland, Distributions of Statistics over Pattern-Avoiding Permutations, arXiv preprint arXiv:1812.07112 [math.CO], 2018.

Crossrefs

Cf. A109447. See A054142 for another version.

Programs

Maple

T:= (n, k)-> binomial(n, 2*k+irem(n, 2)):
seq(seq(T(n, k), k=0..floor(n/2)), n=0..20);  # Alois P. Heinz, Feb 07 2014

Mathematica

Flatten[ Table[ If[ EvenQ[n - k], Binomial[n, k], {}], {n, 0, 15}, {k, 0, n}]] (* Robert G. Wilson v *)

Extensions

More terms from Robert G. Wilson v, Aug 30 2005