OFFSET
0,4
COMMENTS
See A193722 for the definition of fusion of two sequences of polynomials or triangular arrays.
Triangle T(n,k), read by rows, given by [1,1,0,0,0,0,0,0,0,...] DELTA [1,2,0,0,0,0,0,0,0,...] where DELTA is the operator defined in A084938. - Philippe Deléham, Oct 04 2011
FORMULA
T(n,k) = 3*T(n-1,k-1) + 2*T(n-1,k) with T(0,0)=T(1,0)=T(1,1)=1. - Philippe Deléham, Oct 05 2011
G.f.: (-1+x+2*x*y)/(-1+2*x+3*x*y). - R. J. Mathar, Aug 11 2015
EXAMPLE
First six rows:
1;
1, 1;
2, 5, 3;
4, 16, 21, 9;
8, 44, 90, 81, 27;
16, 112, 312, 432, 297, 81;
MATHEMATICA
z = 8; a = 1; b = 2; c = 1; d = 1;
p[n_, x_] := (a*x + b)^n ; q[n_, x_] := (c*x + d)^n
t[n_, k_] := Coefficient[p[n, x], x^k]; t[n_, 0] := p[n, x] /. x -> 0;
w[n_, x_] := Sum[t[n, k]*q[n + 1 - k, x], {k, 0, n}]; w[-1, x_] := 1
g[n_] := CoefficientList[w[n, x], {x}]
TableForm[Table[Reverse[g[n]], {n, -1, z}]]
Flatten[Table[Reverse[g[n]], {n, -1, z}]] (* A193724 *)
TableForm[Table[g[n], {n, -1, z}]]
Flatten[Table[g[n], {n, -1, z}]] (* A193725 *)
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Clark Kimberling, Aug 04 2011
STATUS
approved