A193790 Triangular array: the fusion of polynomial sequences P and Q given by p(n,x)=(2x+1)^n and q(n,x)=1+x^n.

1, 1, 1, 1, 2, 3, 1, 4, 4, 9, 1, 6, 12, 8, 27, 1, 8, 24, 32, 16, 81, 1, 10, 40, 80, 80, 32, 243, 1, 12, 60, 160, 240, 192, 64, 729, 1, 14, 84, 280, 560, 672, 448, 128, 2187, 1, 16, 112, 448, 1120, 1792, 1792, 1024, 256, 6561, 1, 18, 144, 672, 2016, 4032, 5376

Offset: 0

Author

Clark Kimberling, Aug 05 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
1....2....3
1....4....4....9
1....6....12....8...27
1....8....24....32...16...81
		

Crossrefs

Cf. A193791.

Programs

Mathematica

z = 10; a = 2; b = 1;
p[n_, x_] := (a*x + b)^n
q[n_, x_] := 1 + x^n ; 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}]]  (* A193790 *)
TableForm[Table[g[n], {n, -1, z}]]
Flatten[Table[g[n], {n, -1, z}]]  (* A193791 *)