A193823 Triangular array: the fusion of polynomial sequences P and Q given by p(n,x)=(2x+1)^n and q(n,x)=x^n+x^(n-1)+...+x+1.

1, 1, 1, 1, 3, 3, 1, 5, 9, 9, 1, 7, 19, 27, 27, 1, 9, 33, 65, 81, 81, 1, 11, 51, 131, 211, 243, 243, 1, 13, 73, 233, 473, 665, 729, 729, 1, 15, 99, 379, 939, 1611, 2059, 2187, 2187, 1, 17, 129, 577, 1697, 3489, 5281, 6305, 6561, 6561, 1, 19, 163, 835, 2851

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
1....3....3
1....5....9....9
1....7....19...27...27
1....9....33...65...81...81
		

Crossrefs

Cf. A193722, A193804. A119258, A193860.

Programs

Mathematica

z = 10; a = 2; b = 1;
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}]]   (* A193823 *)
TableForm[Table[g[n], {n, -1, z}]]
Flatten[Table[g[n], {n, -1, z}]]  (* A193824 *)

Formula

From Peter Bala, Jul 16 2013: (Start)

T(n,k) = sum {i = 0..k} binomial(n-1,k-i)*2^(k-i) for 0 <= k <= n.

O.g.f.: (1 - 2*x*t)^2/( (1 - 3*x*t)*(1 - (2*x + 1)*t) ) = 1 + (1 + x)*t + (1 + 3*x + 3*x^2)*t^2 + .... Cf. A193860.

For n >= 1, the n-th row polynomial R(n,x) = 1/(x-1)*( 3^(n-1)*x^(n+1) - (2*x + 1)^(n-1) ). (End)