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

0,5

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

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.

Example

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; ....

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_ *)

Crossrefs

Cf. A109447. See A054142 for another version.

Sequence in context: A115017 A088439 A162315 * A088441 A061857 A275464

Adjacent sequences: A109443 A109444 A109445 * A109447 A109448 A109449

Keyword

easy,nonn,tabf

Author

_Philippe Deléham_, Aug 27 2005

Extensions

More terms from _Robert G. Wilson v_, Aug 30 2005

Status

approved