%I #21 Dec 19 2018 14:27:40

%S 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,

%T 36,1,1,45,210,210,45,1,11,165,462,330,55,1,1,66,495,924,495,66,1,13,

%U 286,1287,1716,715,78,1,1,91,1001,3003,3003,1001,91,1,15,455,3003,6435,5005

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

%C 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

%H Alois P. Heinz, <a href="/A109446/b109446.txt">Rows n = 0..200, flattened</a>

%H M. Bukata, R. Kulwicki, N. Lewandowski, L. Pudwell, J. Roth, and T. Wheeland, <a href="https://arxiv.org/abs/1812.07112">Distributions of Statistics over Pattern-Avoiding Permutations</a>, arXiv preprint arXiv:1812.07112 [math.CO], 2018.

%e Starred terms in Pascal's triangle (A007318), read by rows:

%e 1*;

%e 1, 1*;

%e 1*, 2, 1*;

%e 1, 3*, 3, 1*;

%e 1*, 4, 6*, 4, 1*;

%e 1, 5*, 10, 10*, 5, 1*;

%e 1*, 6, 15*, 20, 15*, 6, 1*;

%e 1, 7*, 21, 35*, 35, 21*, 7, 1*;

%e 1*, 8, 28*, 56, 70*, 56, 28*, 8, 1*;

%e 1, 9*, 36, 84*, 126, 126*, 84, 36*, 9, 1*;

%e Rows in A086645 (1; 1, 1; 1, 6, 1; ...) interspersed with rows in A103327 (1; 3, 1; 5, 10, 1; ...).

%e 1; 1; 1, 1; 3, 1; 1, 6, 1; 5, 10, 1; 1, 15, 15, 1; 7, 35, 21, 1; ....

%p T:= (n, k)-> binomial(n, 2*k+irem(n, 2)):

%p seq(seq(T(n, k), k=0..floor(n/2)), n=0..20); # _Alois P. Heinz_, Feb 07 2014

%t Flatten[ Table[ If[ EvenQ[n - k], Binomial[n, k], {}], {n, 0, 15}, {k, 0, n}]] (* _Robert G. Wilson v_ *)

%Y Cf. A109447. See A054142 for another version.

%K easy,nonn,tabf

%O 0,5

%A _Philippe Deléham_, Aug 27 2005

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