A116088 - OEIS

%I #13 Sep 08 2022 08:45:24

%S 1,0,1,0,2,1,0,1,4,1,0,0,6,6,1,0,0,4,15,8,1,0,0,1,20,28,10,1,0,0,0,15,

%T 56,45,12,1,0,0,0,6,70,120,66,14,1,0,0,0,1,56,210,220,91,16,1,0,0,0,0,

%U 28,252,495,364,120,18,1

%N Riordan array (1, x*(1+x)^2).

%H Michael De Vlieger, <a href="/A116088/b116088.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)^2).

%F Number triangle T(n,k) = C(2*k, n-k) = C(n,k)*C(3*k,n)/C(3*k,k).

%e Triangle begins as:

%e   1;

%e   0, 1;

%e   0, 2, 1;

%e   0, 1, 4,  1;

%e   0, 0, 6,  6,  1;

%e   0, 0, 4, 15,  8,  1;

%e   0, 0, 1, 20, 28, 10,  1;

%e   0, 0, 0, 15, 56, 45, 12, 1;

%t Flatten[Table[Binomial[2k,n-k],{n,0,20},{k,0,n}]] (* _Harvey P. Dale_, Oct 22 2012 *)

%o (PARI) {T(n,k) = binomial(2*k, n-k)}; \\ _G. C. Greubel_, May 09 2019

%o (Magma) [[Binomial(2*k, n-k): k in [0..n]]: n in [0..10]]; // _G. C. Greubel_, May 09 2019

%o (Sage) [[binomial(2*k, n-k) for k in (0..n)] for n in (0..10)] # _G. C. Greubel_, May 09 2019

%o (GAP) Flat(List([0..10], n->List([0..n], k-> Binomial(2*k, n-k) ))); # _G. C. Greubel_, May 09 2019

%Y Row sums are A002478. Diagonal sums are A094686. Inverse is (-1)^(n-k) * A109971(n,k). Unsigned version of A109970.

%K easy,nonn,tabl

%O 0,5

%A _Paul Barry_, Feb 04 2006