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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
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
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 July 17 10:03 EDT 2024. Contains 374375 sequences. (Running on oeis4.)