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A109446 Binomial coefficients C(n,k) with n-k even, read by rows. 2
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 (list; graph; refs; listen; history; text; internal format)
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

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Last modified August 11 15:10 EDT 2022. Contains 356066 sequences. (Running on oeis4.)