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Riordan array (1, x*(1+x)^3).
3

%I #15 Sep 08 2022 08:45:24

%S 1,0,1,0,3,1,0,3,6,1,0,1,15,9,1,0,0,20,36,12,1,0,0,15,84,66,15,1,0,0,

%T 6,126,220,105,18,1,0,0,1,126,495,455,153,21,1,0,0,0,84,792,1365,816,

%U 210,24,1,0,0,0,36,924,3003,3060,1330,276,27,1

%N Riordan array (1, x*(1+x)^3).

%H Michael De Vlieger, <a href="/A116089/b116089.txt">Table of n, a(n) for n = 0..11475</a> (Rows 0 <= n <= 150).

%H Milan Janjić, <a href="https://www.emis.de/journals/JIS/VOL21/Janjic2/janjic103.html">Pascal Matrices and Restricted Words</a>, J. Int. Seq., Vol. 21 (2018), Article 18.5.2.

%F G.f.: 1/(1-x*y*(1+x)^3).

%F Number triangle T(n,k) = C(3*k,n-k) = C(n,k)*C(4*k,n)/C(4*k,k).

%e Triangle begins as:

%e 1;

%e 0, 1;

%e 0, 3, 1;

%e 0, 3, 6, 1;

%e 0, 1, 15, 9, 1;

%e 0, 0, 20, 36, 12, 1;

%e 0, 0, 15, 84, 66, 15, 1;

%e 0, 0, 6, 126, 220, 105, 18, 1;

%e 0, 0, 1, 126, 495, 455, 153, 21, 1;

%e 0, 0, 0, 84, 792, 1365, 816, 210, 24, 1;

%e 0, 0, 0, 36, 924, 3003, 3060, 1330, 276, 27, 1;

%t Flatten[Table[Binomial[3k,n-k],{n,0,20},{k,0,n}]] (* _Harvey P. Dale_, Feb 05 2012 *)

%o (PARI) {T(n,k) = binomial(3*k, n-k)}; \\ _G. C. Greubel_, May 09 2019

%o (Magma) [[Binomial(3*k, n-k): k in [0..n]]: n in [0..12]]; // _G. C. Greubel_, May 09 2019

%o (Sage) [[binomial(3*k, n-k) for k in (0..n)] for n in (0..12)] # _G. C. Greubel_, May 09 2019

%o (GAP) Flat(List([0..12], n-> List([0..n], k-> Binomial(3*k, n-k) ))); # _G. C. Greubel_, May 09 2019

%Y Row sums are A099234. Diagonal sums are A116090.

%K easy,nonn,tabl

%O 0,5

%A _Paul Barry_, Feb 04 2006