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A109446 Binomial coefficients C(n,k) with n-k even, read by rows.

%I A109446 #21 Dec 19 2018 14:27:40
%S A109446 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 A109446 36,1,1,45,210,210,45,1,11,165,462,330,55,1,1,66,495,924,495,66,1,13,
%U A109446 286,1287,1716,715,78,1,1,91,1001,3003,3003,1001,91,1,15,455,3003,6435,5005
%N A109446 Binomial coefficients C(n,k) with n-k even, read by rows.
%C A109446 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 A109446 Alois P. Heinz, <a href="/A109446/b109446.txt">Rows n = 0..200, flattened</a>
%H A109446 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 A109446 Starred terms in Pascal's triangle (A007318), read by rows:
%e A109446   1*;
%e A109446   1,  1*;
%e A109446   1*,  2,  1*;
%e A109446   1,  3*,   3,  1*;
%e A109446   1*,  4,  6*,   4,  1*;
%e A109446   1,  5*,  10, 10*,   5,   1*;
%e A109446   1*,  6, 15*,  20, 15*,    6,  1*;
%e A109446   1,  7*,  21, 35*,  35,  21*,   7,  1*;
%e A109446   1*,  8, 28*,  56, 70*,   56, 28*,   8, 1*;
%e A109446   1,  9*,  36, 84*, 126, 126*,  84, 36*,  9, 1*;
%e A109446 Rows in A086645 (1; 1, 1; 1, 6, 1; ...) interspersed with rows in A103327 (1; 3, 1; 5, 10, 1; ...).
%e A109446 1; 1; 1, 1; 3, 1; 1, 6, 1; 5, 10, 1; 1, 15, 15, 1; 7, 35, 21, 1; ....
%p A109446 T:= (n, k)-> binomial(n, 2*k+irem(n, 2)):
%p A109446 seq(seq(T(n, k), k=0..floor(n/2)), n=0..20);  # _Alois P. Heinz_, Feb 07 2014
%t A109446 Flatten[ Table[ If[ EvenQ[n - k], Binomial[n, k], {}], {n, 0, 15}, {k, 0, n}]] (* _Robert G. Wilson v_ *)
%Y A109446 Cf. A109447. See A054142 for another version.
%K A109446 easy,nonn,tabf
%O A109446 0,5
%A A109446 _Philippe Deléham_, Aug 27 2005
%E A109446 More terms from _Robert G. Wilson v_, Aug 30 2005