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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+1)^n.
2

%I #8 Dec 26 2023 09:54:00

%S 1,2,3,3,8,6,5,18,23,11,8,37,66,55,19,13,73,167,196,120,32,21,139,388,

%T 587,511,246,53,34,259,853,1578,1777,1225,484,87,55,474,1799,3933,

%U 5428,4857,2765,924,142,89,856,3678,9275,15147,16642,12333,5969

%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+1)^n.

%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....3

%e 3....8....6

%e 5....18...23....11

%e 8....37...66....55....19

%e 13...73...167...196...120...32

%t z = 11;

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

%t q[n_, x_] := (x + 1)^n;

%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}]] (* A193997 *)

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

%t Flatten[Table[h[n], {n, -1, z}]] (* A193998 *)

%Y Cf. A193842, A193998.

%K nonn,tabl

%O 0,2

%A _Clark Kimberling_, Aug 11 2011