A193821 Triangular array: the fusion of polynomial sequences P and Q given by p(n,x)=(x+2)^n and q(n,x)=x^n+x^(n-1)+...+x+1.

1, 1, 1, 2, 3, 3, 4, 8, 9, 9, 8, 20, 26, 27, 27, 16, 48, 72, 80, 81, 81, 32, 112, 192, 232, 242, 243, 243, 64, 256, 496, 656, 716, 728, 729, 729, 128, 576, 1248, 1808, 2088, 2172, 2186, 2187, 2187, 256, 1280, 3072, 4864, 5984, 6432, 6544, 6560, 6561, 6561

Offset: 0

Author

Clark Kimberling, Aug 06 2011

Keywords

• nonn
• tabl

Comments

See A193722 for the definition of fusion of two sequences of polynomials or triangular arrays.

Examples

			First six rows:
1
1....1
2....3....3
4....8....9....9
8....20...26...27...27
16...48...72...80...81...81
		

Crossrefs

Cf. A193722, A193822.

Programs

Mathematica

p[n_, x_] := (a*x + b)^n
q[0, x_] := 1
q[n_, x_] := x*q[n - 1, x] + 1; q[n_, 0] := q[n, x] /. x -> 0;
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}]]   (* A193821 *)
TableForm[Table[g[n], {n, -1, z}]]
Flatten[Table[g[n], {n, -1, z}]]  (* A193822 *)