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Triangular array: the fission 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)=x*q(n-1,x)+n+1, with q(0,x)=1.
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%I #7 Mar 30 2012 18:57:39

%S 1,2,5,3,8,14,5,13,23,34,8,21,37,55,74,13,34,60,89,120,152,21,55,97,

%T 144,194,246,299,34,89,157,233,314,398,484,571,55,144,254,377,508,644,

%U 783,924,1066,89,233,411,610,822,1042,1267,1495,1725,1956,144,377

%N Triangular array: the fission 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)=x*q(n-1,x)+n+1, with q(0,x)=1.

%C See A193842 for the definition of the fission of P by Q, where P and Q are sequences of polynomials or triangular arrays (of coefficients of polynomials).

%e First six rows:

%e 1

%e 2....5

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

%e 5....13...23...34

%e 8....21...37...55...74

%e 13...34...60...89...120...152

%t z = 11;

%t p[n_, x_] := Sum[Fibonacci[k + 1]*x^(n - k), {k, 0, n}];

%t q[n_, x_] := x*q[n - 1, x] + n + 1; q[0, n_] := 1;

%t p1[n_, k_] := Coefficient[p[n, x], x^k];

%t p1[n_, 0] := p[n, x] /. x -> 0;

%t d[n_, x_] := Sum[p1[n, k]*q[n - 1 - k, x], {k, 0, n - 1}]

%t h[n_] := CoefficientList[d[n, x], {x}]

%t TableForm[Table[Reverse[h[n]], {n, 0, z}]]

%t Flatten[Table[Reverse[h[n]], {n, -1, z}]] (* A194007 *)

%t TableForm[Table[h[n], {n, 0, z}]]

%t Flatten[Table[h[n], {n, -1, z}]] (* A194008 *)

%Y Cf. A193842, A194008.

%K nonn,tabl

%O 0,2

%A _Clark Kimberling_, Aug 11 2011