A172249 - OEIS

%I #17 Nov 15 2020 12:59:19

%S 1,0,3,0,1,8,0,0,6,21,0,0,1,25,55,0,0,0,9,90,144,0,0,0,1,51,300,377,0,

%T 0,0,0,12,234,954,987,0,0,0,0,1,86,951,2939,2584,0,0,0,0,0,15,480,

%U 3573,8850,6765,0,0,0,0,0,1,130,2305,12707,26195,17711,0,0,0,0,0,0,18,855

%N Triangle, read by rows, given by [0,1/3,-1/3,0,0,0,0,0,0,0,...] DELTA [3,-1/3,1/3,0,0,0,0,0,0,0,...] where DELTA is the operator defined in A084938.

%C Diagonal sums : |A077897|. Column sums : A001353 .

%F T(n,k) = 3*T(n-1,k-1) + T(n-2,k-1) - T(n-2,k-2), T(0,0)=1, T(n,k) = 0 if k>n or if k<0.

%F Sum_{k, 0<=k<=n} T(n,k)= 3^n = A000244(n) (row sums).

%F G.f.: 1/(1-3*x*y-x^2*y+x^2*y^2). - _R. J. Mathar_, Aug 11 2015

%F T(n,k) = 2*Sum_{j=1..n+k} j*C(n+j,2*n-2*k+2*j)*C(n-k+j,j)/(n+j), T(0,0)=1. - _Vladimir Kruchinin_, Oct 28 2020

%e Triangle begins :

%e 1,

%e 0,3,

%e 0,1,8,

%e 0,0,6,21,

%e 0,0,1,25,55,

%e 0,0,0,9,90,144,

%e 0,0,0,1,51,300,377,

%e 0,0,0,0,12,234,954,987,

%e 0,0,0,0,1,86,951,2939,2584,

%e 0,0,0,0,0,15,480,3573,8850,6765,

%e 0,0,0,0,0,1,130,2305,12707,26195,17711,

%o (Maxima)

%o T(n,k):=2*sum((j*binomial(n+j,2*n-2*k+2*j)*binomial(n-k+j,j))/(n+j),j,1,n+k); /* Vladimir Kruchinin_, Oct 28 2020 */

%Y Cf. A001871, A001906, A125662.

%K nonn,tabl

%O 0,3

%A _Philippe Deléham_, Jan 29 2010