A193724 Triangular array: the fusion of polynomial sequences P and Q given by p(n,x)=(x+2)^n and q(n,x)=(x+1)^n.

1, 1, 1, 2, 5, 3, 4, 16, 21, 9, 8, 44, 90, 81, 27, 16, 112, 312, 432, 297, 81, 32, 272, 960, 1800, 1890, 1053, 243, 64, 640, 2736, 6480, 9180, 7776, 3645, 729, 128, 1472, 7392, 21168, 37800, 43092, 30618, 12393, 2187, 256, 3328, 19200, 64512, 139104

Offset: 0

Author

Clark Kimberling, Aug 04 2011

Keywords

• nonn
• tabl

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

Examples

			First six rows:
   1;
   1,   1;
   2,   5,   3;
   4,  16,  21,   9;
   8,  44,  90,  81,  27;
  16, 112, 312, 432, 297, 81;
		

Crossrefs

Cf. A084938, A193722, A193725.

Programs

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 *)

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