A193915 Triangular array: the fusion of (p(n,x)) by (q(n,x)), where p(n,x)=sum{F(k+1)*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, 4, 4, 2, 16, 12, 8, 4, 48, 40, 24, 14, 7, 160, 128, 80, 44, 24, 12, 512, 416, 256, 144, 76, 40, 20, 1664, 1344, 832, 464, 248, 128, 66, 33, 5376, 4352, 2688, 1504, 800, 416, 212, 108, 54, 17408, 14080, 8704, 4864, 2592, 1344, 688, 348, 176, 88

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...1

1...1...2

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

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

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..54.

Example

First six rows:
1
2....1
4....4....2
16...12...8...4
48...40...24..14..7
160..128..80..44..24..12

Mathematica

z = 12;
p[n_, x_] := Sum[Fibonacci[k + 1]*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}]] (* A193915 *)
TableForm[Table[g[n], {n, -1, z}]]
Flatten[Table[g[n], {n, -1, z}]]  (* A193916 *)

Crossrefs

Cf. A193722, A193916, A193908.

Sequence in context: A375049 A129159 A095830 * A101621 A086484 A349572

Adjacent sequences: A193912 A193913 A193914 * A193916 A193917 A193918

Keyword

nonn,tabl

Author

Clark Kimberling, Aug 09 2011

Status

approved