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A193967
Triangular array: the fusion of (p(n,x)) by (q(n,x)), where p(n,x)=sum{L(k+1)*x^(n-k) : 0<=k<=n}, where F=A000032 (Lucas numbers), and q(n,x)=sum{F(k+1)*x^(n-k) : 0<=k<=n}, where F=A000045 (Fibonacci numbers).
2
1, 1, 1, 3, 4, 7, 4, 7, 12, 19, 7, 11, 21, 33, 54, 11, 18, 33, 54, 88, 142, 18, 29, 54, 87, 144, 232, 376, 29, 47, 87, 141, 232, 376, 609, 985, 47, 76, 141, 228, 376, 608, 987, 1596, 2583, 76, 123, 228, 369, 608, 984, 1596, 2583, 4180, 6763, 123, 199, 369
OFFSET
0,4
COMMENTS
See A193722 for the definition of fusion of two sequences of polynomials or triangular arrays.
EXAMPLE
First six rows:
1
1...1
3...4...7
4...7...12..19
7...11..21..33..54
11..18..33..54..88..142
MATHEMATICA
z = 12;
p[n_, x_] := Sum[LucasL[k + 1]*x^(n - k), {k, 0, n}];
q[n_, x_] := Sum[Fibonacci[k + 1]*x^(n - k), {k, 0, n}];
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}]] (* A193967 *)
TableForm[Table[g[n], {n, -1, z}]]
Flatten[Table[g[n], {n, -1, z}]] (* A193968 *)
CROSSREFS
Sequence in context: A200681 A161775 A282535 * A109823 A337124 A267447
KEYWORD
nonn,tabl
AUTHOR
Clark Kimberling, Aug 10 2011
STATUS
approved