Milan Janjic, <a href="httphttps://www.emiscs.amsuwaterloo.orgca/journals/JIS/VOL19/Janjic/janjic73.html">Binomial Coefficients and Enumeration of Restricted Words</a>, Journal of Integer Sequences, 2016, Vol 19, #16.7.3

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Milan Janjic, <a href="http://www.emis.ams.org/journals/JIS/VOL19/Janjic/janjic73.html">Binomial Coefficients and Enumeration of Restricted Words</a>, Journal of Integer Sequences, 2016, Vol 19, #16.7.3

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Read as a number triangle, this is the Riordan array (1, x(1+x+x^2)) with T(n,k) =sum Sum_{i=0..floor((n+k)/2), } C(k,2i+2k-n)*C(2i+2k-n,i)}. Rows start {1}, {0,1}, {0,1,1}, {0,1,2,1}, {0,0,3,3,1},... Row sums are then the trinomial numbers A000073(n+2). Diagonal sums are A013979. - Paul Barry, Feb 15 2005

G. C. Greubel, <a href="/A071675/b071675.txt">Table of n, a(n) for the first 50 rows, flattened</a>

T(n, k) = T(n-1, k) + T(n-1, k-1) + T(n-1, k-2) starting with T(0, 0)=1. See A027907 for more.

As a number triangle, T(n, k) =sum Sum_{i=0..floor((n-k)/2), } C(n-k-i, i) * C(k, n-k-i)}; . - Paul Barry, Apr 26 2005

T[n_, k_] := Sum[Binomial[n - k - j, j]*Binomial[k, n - k - j], {j, 0,

Floor[(n - k)/2]}]; Table[T[n, k], {n, 0, 10}, {k, 0, n}] // Flatten (* G. C. Greubel, Feb 28 2017 *)

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Let {a_(k,i)}, k>=1, i=0,...,k, be the k-th antidiagonal of the array. Then s_k(n)=sum{i=0,...,k}a_(k,i)* binomial(n,k) is the n-th element of the k-th column of A213742. For example, s_1(n)=binomial(n,1)=n is the first column of A213742 for n>1, s_2(n)=binomial(n,1)+binomial(n,2)is the second column of A213742 for n>1, etc. In particular (see comment in A213742) in cases k=4,5,6,7,8, s_k(n) is A005718(n+2), A005719(n), A005720(n), A001919(n), A064055(n+3), respectively. - _Vladimir Shevelev _ and _Peter J. C. Moses, _, Jun 22 2012

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