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A194009 Triangular array: the fission of (p(n,x)) by (q(n,x)), where p(n,x)=x*p(n-1,x)+n+1 with p(0,x)=1, and q(n,x)=sum{F(k+1)*x^(n-k) : 0<=k<=n}, where F=A000045 (Fibonacci numbers).

%I A194009 #7 Jan 09 2024 12:29:17
%S A194009 2,3,5,4,7,13,5,9,17,28,6,11,21,35,58,7,13,25,42,70,114,8,15,29,49,82,
%T A194009 134,218,9,17,33,56,94,154,251,407,10,19,37,63,106,174,284,461,747,11,
%U A194009 21,41,70,118,194,317,515,835,1352,12,23,45,77,130,214,350,569
%N A194009 Triangular array:  the fission of (p(n,x)) by (q(n,x)), where p(n,x)=x*p(n-1,x)+n+1 with p(0,x)=1, and q(n,x)=sum{F(k+1)*x^(n-k) : 0<=k<=n}, where F=A000045 (Fibonacci numbers).
%C A194009 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 A194009 First six rows:
%e A194009 2
%e A194009 3...5
%e A194009 4...7....13
%e A194009 5...9....17...28
%e A194009 6...11...21...35...58
%e A194009 7...13...25...42...70...114
%t A194009 z = 11;
%t A194009 p[n_, x_] := x*p[n - 1, x] + n + 1; p[0, n_] := 1;
%t A194009 q[n_, x_] := Sum[Fibonacci[k + 1]*x^(n - k), {k, 0, n}];
%t A194009 p1[n_, k_] := Coefficient[p[n, x], x^k];
%t A194009 p1[n_, 0] := p[n, x] /. x -> 0;
%t A194009 d[n_, x_] := Sum[p1[n, k]*q[n - 1 - k, x], {k, 0, n - 1}]
%t A194009 h[n_] := CoefficientList[d[n, x], {x}]
%t A194009 TableForm[Table[Reverse[h[n]], {n, 0, z}]]
%t A194009 Flatten[Table[Reverse[h[n]], {n, -1, z}]]  (* A194009 *)
%t A194009 TableForm[Table[h[n], {n, 0, z}]]
%t A194009 Flatten[Table[h[n], {n, -1, z}]]  (* A194010 *)
%Y A194009 Cf. A193842, A194010.
%K A194009 nonn,tabl
%O A194009 0,1
%A A194009 _Clark Kimberling_, Aug 11 2011