|
%I #5 Mar 30 2012 18:57:39
%S 1,1,3,1,4,7,2,7,12,21,3,11,19,33,54,5,18,31,54,88,144,8,29,50,87,142,
%T 232,376,13,47,81,141,230,376,609,987,21,76,131,228,372,608,985,1596,
%U 2583,34,123,212,369,602,984,1594,2583,4180,6765,55,199,343,597
%N 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)=sum{L(k+1)*x^(n-k) : 0<=k<=n}, where F=A000032 (Lucas 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...3
%e 1...4...7
%e 2...7...12...21
%e 3...11..19...33...54
%e 5...18..31...54...88...144
%t z = 12;
%t p[n_, x_] := Sum[Fibonacci[k + 1]*x^(n - k), {k, 0, n}];
%t q[n_, x_] := Sum[LucasL[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}]] (* A193969 *)
%t TableForm[Table[g[n], {n, -1, z}]]
%t Flatten[Table[g[n], {n, -1, z}]] (* A193970 *)
%Y Cf. A193722, A193970.
%K nonn,tabl
%O 0,3
%A _Clark Kimberling_, Aug 10 2011
|
