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A119301 Triangle read by rows: T(n,k) = binomial(3*n-k,n-k).

%I A119301 #27 Jun 11 2024 09:39:11
%S A119301 1,3,1,15,5,1,84,28,7,1,495,165,45,9,1,3003,1001,286,66,11,1,18564,
%T A119301 6188,1820,455,91,13,1,116280,38760,11628,3060,680,120,15,1,735471,
%U A119301 245157,74613,20349,4845,969,153,17,1,4686825,1562275,480700,134596,33649,7315
%N A119301 Triangle read by rows: T(n,k) = binomial(3*n-k,n-k).
%C A119301 First column is A005809. Second column is A025174.
%C A119301 Row sums are A045721. Inverse is Riordan array (1-3x,x(1-x)^2), A119302.
%F A119301 G.f. g(x) = 2*sin(arcsin(3*sqrt(3*x)/2)/3)/sqrt(3*x) satisfies g(x) = 1/(1-x*g(x)^2).
%F A119301 Riordan array (1/(1-3*x*g(x)^2),x*g(x)^2) where g(x)=1+x*g(x)^3.
%F A119301 'Horizontal' recurrence equation: T(n,0) = binomial(3*n,n) and for k >= 1, T(n,k) = Sum_{i = 1..n+1-k} i*T(n-1,k-2+i). - _Peter Bala_, Dec 28 2014
%F A119301 T(n, k) = Sum_{j = 0..n} binomial(n+j-1, j)*binomial(2*n-k-j, n). - _Peter Bala_, Jun 04 2024
%e A119301 Triangle begins
%e A119301   1,
%e A119301   3, 1,
%e A119301   15, 5, 1,
%e A119301   84, 28, 7, 1,
%e A119301   495, 165, 45, 9, 1,
%e A119301   3003, 1001, 286, 66, 11, 1,
%e A119301   18564, 6188, 1820, 455, 91, 13, 1,
%e A119301   116280, 38760, 11628, 3060, 680, 120, 15, 1
%e A119301   ...
%e A119301 Horizontal recurrence: T(4,1) = 1*84 + 2*28 + 3*7 + 4*1 = 165. - _Peter Bala_, Dec 29 2014
%p A119301 T := proc(n,k) option remember;
%p A119301 `if`(n = 0, 1, add(i*T(n-1,k-2+i),i=1..n+1-k)) end:
%p A119301 for n from 0 to 9 do print(seq(T(n,k),k=0..n)) od; # _Peter Luschny_, Dec 30 2014
%t A119301 Flatten[Table[Binomial[3n-k,n-k],{n,0,10},{k,0,n}]] (* _Harvey P. Dale_, Jul 28 2012 *)
%Y A119301 Cf. A092392, A119304.
%Y A119301 Cf. A005809, A025174, A045721, A119302.
%K A119301 easy,nonn,tabl
%O A119301 0,2
%A A119301 _Paul Barry_, May 13 2006