|
%I #6 Mar 30 2012 18:57:39
%S 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,
%T 144,232,376,29,47,87,141,232,376,609,985,47,76,141,228,376,608,987,
%U 1596,2583,76,123,228,369,608,984,1596,2583,4180,6763,123,199,369
%N 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).
%C See A193722 for the definition of fusion of two sequences of polynomials or triangular arrays.
%e First six rows:
%e 1
%e 1...1
%e 3...4...7
%e 4...7...12..19
%e 7...11..21..33..54
%e 11..18..33..54..88..142
%t z = 12;
%t p[n_, x_] := Sum[LucasL[k + 1]*x^(n - k), {k, 0, n}];
%t q[n_, x_] := Sum[Fibonacci[k + 1]*x^(n - k), {k, 0, n}];
%t t[n_, k_] := Coefficient[p[n, x], x^k]; t[n_, 0] := p[n, x] /. x -> 0;
%t w[n_, x_] := Sum[t[n, k]*q[n + 1 - k, x], {k, 0, n}]; w[-1, x_] := 1
%t g[n_] := CoefficientList[w[n, x], {x}]
%t TableForm[Table[Reverse[g[n]], {n, -1, z}]]
%t Flatten[Table[Reverse[g[n]], {n, -1, z}]] (* A193967 *)
%t TableForm[Table[g[n], {n, -1, z}]]
%t Flatten[Table[g[n], {n, -1, z}]] (* A193968 *)
%Y Cf. A193722, A193968.
%K nonn,tabl
%O 0,4
%A _Clark Kimberling_, Aug 10 2011
|
