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.
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, 0, 0, 0, 12, 234, 954, 987, 0, 0, 0, 0, 1, 86, 951, 2939, 2584, 0, 0, 0, 0, 0, 15, 480, 3573, 8850, 6765, 0, 0, 0, 0, 0, 1, 130, 2305, 12707, 26195, 17711, 0, 0, 0, 0, 0, 0, 18, 855
Offset: 0
Examples
Triangle begins : 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, 0,0,0,0,12,234,954,987, 0,0,0,0,1,86,951,2939,2584, 0,0,0,0,0,15,480,3573,8850,6765, 0,0,0,0,0,1,130,2305,12707,26195,17711,
Programs
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Maxima
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 */
Formula
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.
Sum_{k, 0<=k<=n} T(n,k)= 3^n = A000244(n) (row sums).
G.f.: 1/(1-3*x*y-x^2*y+x^2*y^2). - R. J. Mathar, Aug 11 2015
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
Comments