A109446 - OEIS

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

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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