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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.
2
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
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
KEYWORD
nonn,tabl
AUTHOR
Clark Kimberling, Aug 09 2011
STATUS
approved