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

0,3

Comments

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

Links

Table of n, a(n) for n=0..73.

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

Example

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,

Prog

(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 */

Crossrefs

Cf. A001871, A001906, A125662.

Sequence in context: A216806 A290776 A373086 * A208758 A320161 A373637

Adjacent sequences: A172246 A172247 A172248 * A172250 A172251 A172252

Keyword

nonn,tabl

Author

Philippe Deléham, Jan 29 2010

Status

approved