OFFSET
0,3
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 (1,0,-1,1,0,0,0,0,0,0,0,...) DELTA (2,1,-2,2,0,0,0,0,0,0,0,...) where DELTA is the operator defined in A084938. - Philippe Deléham, Oct 05 2011
FORMULA
T(n,k) = T(n-1,k) + 3*T(n-1,k-1) - T(n-2,k-1) - 2*T(n-2,k-2), T(0,0)=1, T(1,0)=1, T(1,1)=2, T(2,0)=1, T(2,1)=5, T(2,2)=6, T(n,k)=0 if k < 0 or if k > n. - Philippe Deléham, Dec 15 2013
G.f.: (1-x*y+x^2*y+2*x^2*y^2)/((-1+x+2*x*y)*(x*y-1)). - R. J. Mathar, Aug 12 2015
EXAMPLE
First six rows:
1;
1, 2;
1, 5, 6;
1, 7, 17, 14;
1, 9, 31, 49, 30;
1, 11, 49, 111, 129, 62;
MATHEMATICA
z = 10; c = 1; d = 2;
p[0, x_] := 1
p[n_, x_] := x*p[n - 1, x] + 1; p[n_, 0] := p[n, x] /. x -> 0;
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}]] (* A193816 *)
TableForm[Table[g[n], {n, -1, z}]]
Flatten[Table[g[n], {n, -1, z}]] (* A193817 *)
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Clark Kimberling, Aug 06 2011
STATUS
approved