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A172249 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.

%I A172249 #17 Nov 15 2020 12:59:19
%S A172249 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 A172249 0,0,0,12,234,954,987,0,0,0,0,1,86,951,2939,2584,0,0,0,0,0,15,480,
%U A172249 3573,8850,6765,0,0,0,0,0,1,130,2305,12707,26195,17711,0,0,0,0,0,0,18,855
%N A172249 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 A172249 Diagonal sums : |A077897|. Column sums : A001353 .
%F A172249 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 A172249 Sum_{k, 0<=k<=n} T(n,k)= 3^n = A000244(n) (row sums).
%F A172249 G.f.: 1/(1-3*x*y-x^2*y+x^2*y^2). - _R. J. Mathar_, Aug 11 2015
%F A172249 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 A172249 Triangle begins :
%e A172249 1,
%e A172249 0,3,
%e A172249 0,1,8,
%e A172249 0,0,6,21,
%e A172249 0,0,1,25,55,
%e A172249 0,0,0,9,90,144,
%e A172249 0,0,0,1,51,300,377,
%e A172249 0,0,0,0,12,234,954,987,
%e A172249 0,0,0,0,1,86,951,2939,2584,
%e A172249 0,0,0,0,0,15,480,3573,8850,6765,
%e A172249 0,0,0,0,0,1,130,2305,12707,26195,17711,
%o A172249 (Maxima)
%o A172249 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 A172249 Cf. A001871, A001906, A125662.
%K A172249 nonn,tabl
%O A172249 0,3
%A A172249 _Philippe Deléham_, Jan 29 2010