login
Triangle read by rows: T(n,m) = (m/(n-m))*Sum_{k=1..n-m}((-1)^k*binomial(m-1,k-1)*binomial(3*(n-m)-k-1,n-m-k)), T(n,n)=1.
0

%I #22 Apr 15 2016 13:48:48

%S 1,-1,1,-2,-2,1,-7,-3,-3,1,-30,-10,-3,-4,1,-143,-42,-10,-2,-5,1,-728,

%T -198,-42,-8,0,-6,1,-3876,-1001,-198,-35,-5,3,-7,1,-21318,-5304,-1001,

%U -168,-25,-2,7,-8,1,-120175,-29070,-5304,-858,-126,-15,0,12,-9,1

%N Triangle read by rows: T(n,m) = (m/(n-m))*Sum_{k=1..n-m}((-1)^k*binomial(m-1,k-1)*binomial(3*(n-m)-k-1,n-m-k)), T(n,n)=1.

%F G.f.: -1+1/(1-x*y+x*y*(4*sin(asin((3^(3/2)*sqrt(x))/2)/3)^2)/3)

%F T(n,m) = (1/(n-m))*(m*(-1)^(n-m)*Sum_{k=1..n-m} binomial(k-1,n-m-1)*binomial(-2*n+3*m-1,k-1)*binomial(3*n-4*m,n-m-k)), n>m, T(n,n)=1

%e 1;

%e -1,1;

%e -2,-2,1;

%e -7,-3,-3,1;

%e -30,-10,-3,-4,1;

%e -143,-42,-10,-2,-5,1;

%e -728,-198,-42,-8,0,-6,1;

%o (Maxima)

%o T(n,m):=if n=m then 1 else m*(-1)^(n-m)/(n-m)*sum((binomial(k-1,n-m-1)*binomial(-2*n+3*m-1,k-1)*binomial(3*n-4*m,n-m-k)),k,1,n-m);

%Y Cf. A006013.

%K sign,tabl

%O 1,4

%A _Vladimir Kruchinin_, Apr 13 2016