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Triangular array: the self-fusion of (p(n,x)), where p(n,x)=sum{F(k+2)*x^(n-k) : 0<=k<=n}, where F=A000045 (Fibonacci numbers).
2

%I #5 Mar 30 2012 18:57:39

%S 1,1,2,2,5,8,3,8,14,23,5,13,23,39,63,8,21,37,63,103,167,13,34,60,102,

%T 167,272,440,21,55,97,165,270,440,713,1154,34,89,157,267,437,712,1154,

%U 1869,3024,55,144,254,432,707,1152,1867,3024,4894,7919,89,233

%N Triangular array: the self-fusion of (p(n,x)), where p(n,x)=sum{F(k+2)*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. The first five rows of P, from the coefficients of p(n,k):

%C 1

%C 1...2

%C 1...2...3

%C 1...2...3...5

%C 1...2...3...5...8

%e First six rows of A193906:

%e 1

%e 1...2

%e 2...5....8

%e 3...8....14...23

%e 5...13...23...39...63

%e 8...21...37...63...103...167

%t z = 12;

%t p[n_, x_] := Sum[Fibonacci[k + 2]*x^(n - k), {k, 0, n}];

%t q[n_, x_] := p[n, x]

%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}]] (* A193906 *)

%t TableForm[Table[g[n], {n, -1, z}]]

%t Flatten[Table[g[n], {n, -1, z}]] (* A193907 *)

%Y Cf. A193722, A193907.

%K nonn,tabl

%O 0,3

%A _Clark Kimberling_, Aug 08 2011