%I #20 Jul 03 2020 14:55:05

%S 1,0,1,0,1,1,0,1,2,1,0,0,3,3,1,0,0,2,6,4,1,0,0,1,7,10,5,1,0,0,0,6,16,

%T 15,6,1,0,0,0,3,19,30,21,7,1,0,0,0,1,16,45,50,28,8,1,0,0,0,0,10,51,90,

%U 77,36,9,1,0,0,0,0,4,45,126,161,112,45,10,1,0,0,0,0,1,30,141,266,266

%N Array read by antidiagonals of trinomial coefficients.

%C Read as a number triangle, this is the Riordan array (1, x(1+x+x^2)) with T(n,k) = 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

%C 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

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

%H Milan Janjic, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL19/Janjic/janjic73.html">Binomial Coefficients and Enumeration of Restricted Words</a>, Journal of Integer Sequences, 2016, Vol 19, #16.7.3

%F 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.

%F As a number triangle, T(n, k) = Sum_{i=0..floor((n-k)/2)} C(n-k-i, i) * C(k, n-k-i). - _Paul Barry_, Apr 26 2005

%e Rows start

%e 1, 0, 0, 0, 0, 0, ...;

%e 1, 1, 1, 0, 0, 0, 0, ...;

%e 1, 2, 3, 2, 1, 0, 0, ...;

%e 1, 3, 6, 7, 6, 3, 1, 0, ...;

%e 1, 4, 10, 16, 19, 16, 10, 4, 1, ...; etc.

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

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

%Y Visible version of A027907. Row sums are 3^n, i.e. A000244. Central diagonal is A002426. Cf. A071676 for a slight variation.

%K nonn,tabl

%O 0,9

%A _Henry Bottomley_, May 30 2002