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A193997 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.

%I A193997 #8 Dec 26 2023 09:54:00
%S A193997 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 A193997 587,511,246,53,34,259,853,1578,1777,1225,484,87,55,474,1799,3933,
%U A193997 5428,4857,2765,924,142,89,856,3678,9275,15147,16642,12333,5969
%N A193997 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 A193997 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 A193997 First six rows:
%e A193997 1
%e A193997 2....3
%e A193997 3....8....6
%e A193997 5....18...23....11
%e A193997 8....37...66....55....19
%e A193997 13...73...167...196...120...32
%t A193997 z = 11;
%t A193997 p[n_, x_] := Sum[Fibonacci[k + 1]*x^(n - k), {k, 0, n}];
%t A193997 q[n_, x_] := (x + 1)^n;
%t A193997 p1[n_, k_] := Coefficient[p[n, x], x^k];
%t A193997 p1[n_, 0] := p[n, x] /. x -> 0;
%t A193997 d[n_, x_] := Sum[p1[n, k]*q[n - 1 - k, x], {k, 0, n - 1}]
%t A193997 h[n_] := CoefficientList[d[n, x], {x}]
%t A193997 TableForm[Table[Reverse[h[n]], {n, 0, z}]]
%t A193997 Flatten[Table[Reverse[h[n]], {n, -1, z}]]  (* A193997 *)
%t A193997 TableForm[Table[h[n], {n, 0, z}]]
%t A193997 Flatten[Table[h[n], {n, -1, z}]]  (* A193998 *)
%Y A193997 Cf. A193842, A193998.
%K A193997 nonn,tabl
%O A193997 0,2
%A A193997 _Clark Kimberling_, Aug 11 2011