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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.
4
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
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
Sequence in context: A305860 A272983 A195698 * A193904 A154897 A198577
KEYWORD
nonn,tabl
AUTHOR
Clark Kimberling, Aug 09 2011
STATUS
approved