A193908 Triangular array: the fusion of (p(n,x)) by (q(n,x)), where p(n,x)=sum{F(k+2)*x^(n-k) : 0<=k<=n}, where F=A000045 (Fibonacci numbers), and q(n,x)=2x*q(n-1,x)+1 with q(0,x)=1.

1, 2, 1, 8, 6, 3, 24, 20, 12, 6, 80, 64, 40, 22, 11, 256, 208, 128, 72, 38, 19, 832, 672, 416, 232, 124, 64, 32, 2688, 2176, 1344, 752, 400, 208, 106, 53, 8704, 7040, 4352, 2432, 1296, 672, 344, 174, 87, 28160, 22784, 14080, 7872, 4192, 2176, 1112, 564

Offset

0,2

Comments

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

First five rows of P (triangle of coefficients of polynomials p(n,x)):

1

1...2

1...2...3

1...2...3...5

1...2...3...5...8

First five rows of Q:

1

2...1

4...2..1

8...4...2...1

16..8...4...2...1

Links

Table of n, a(n) for n=0..52.

Example

First six rows:
1
2....1
8....6....3
24...20...12...6
80...64...40...22...11
256..208..128..72...38...19

Mathematica

z = 12;
p[n_, x_] := Sum[Fibonacci[k + 2]*x^(n - k), {k, 0, n}];
q[n_, x_] := 2 x*q[n - 1, x] + 1 ; q[0, x_] := 1;
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}]]  (* A193908 *)
TableForm[Table[g[n], {n, -1, z}]]
Flatten[Table[g[n], {n, -1, z}]]  (* A193909 *)

Crossrefs

Cf. A193722, A193909.

Sequence in context: A305860 A272983 A195698 * A193904 A154897 A198577

Adjacent sequences: A193905 A193906 A193907 * A193909 A193910 A193911

Keyword

nonn,tabl

Author

Clark Kimberling, Aug 09 2011

Status

approved