A131129 - OEIS

%I #10 Jan 27 2020 01:35:39

%S 1,1,1,3,4,1,3,9,7,1,3,12,18,10,1,3,15,30,30,13,1,3,18,45,60,45,16,1,

%T 3,21,63,105,105,63,19,1,3,24,84,168,210,168,84,22,1

%N 3*A007318 - 2*A097806, where A007318 = Pascal's triangle and A097806 = the pairwise operator.

%C Row sums = A131128: (1, 2, 8, 20, 44, 92, 188, 380, ...), the binomial transform of (1, 1, 5, 1, 5, 1, 5, ...). Triangle A131108 has row sums (1, 2, 6, 14, 30, 62, ...), the binomial transform of (1, 1, 3, 1, 3, 1, ...). Generalization: Given triangles generated from N*A007318 - (N-1)*A097806, row sums are binomial transforms of (1, 1, (2N-1), 1, (2N-1), 1, ...).

%C Triangle T(n,k), 0 <= k <= n, read by rows given by [1,2,-3,1,0,0,0,0,0,0,0,...] DELTA [1,0,0,1,0,0,0,0,0,0,0,...] where DELTA is the operator defined in A084938. - _Philippe Deléham_, Dec 18 2007

%F G.f.: (1-x*y+2*x^2+2*x^2*y)/((-1+x+x*y)*(x*y-1)). - _R. J. Mathar_, Aug 12 2015

%e First few rows of the triangle:

%e   1;

%e   1,  1;

%e   3,  4,  1;

%e   3,  9,  7,  1;

%e   3, 12, 18, 10,  1;

%e   3, 15, 30, 30, 13,  1;

%e   ...

%Y Cf. A097806, A131128, A095121.

%K nonn,tabl

%O 0,4

%A _Gary W. Adamson_, Jun 16 2007