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A193923 Triangular array: the fusion of (p(n,x)) by (q(n,x)), where p(n,x)=(x+1)^n and q(n,x)=Sum_{k=0..n}F(k+1)*x^(n-k), where F=A000045 (Fibonacci numbers).

%I A193923 #20 Nov 03 2020 11:46:37
%S A193923 1,1,1,1,2,3,1,3,5,8,1,4,8,13,21,1,5,12,21,34,55,1,6,17,33,55,89,144,
%T A193923 1,7,23,50,88,144,233,377,1,8,30,73,138,232,377,610,987,1,9,38,103,
%U A193923 211,370,609,987,1597,2584,1,10,47,141,314,581,979,1596,2584,4181,6765
%N A193923 Triangular array:  the fusion of (p(n,x)) by (q(n,x)), where p(n,x)=(x+1)^n and q(n,x)=Sum_{k=0..n}F(k+1)*x^(n-k), where F=A000045 (Fibonacci numbers).
%C A193923 See A193722 for the definition of fusion of two sequences of polynomials or triangular arrays.
%C A193923 The row sums equal A079289(2*n). - _Johannes W. Meijer_, Aug 12 2013
%H A193923 Michel Marcus, <a href="/A193923/b193923.txt">Rows n=0..100 of triangle, flattened</a>
%H A193923 T. G. Lavers, <a href="https://ajc.maths.uq.edu.au/pdf/10/ocr-ajc-v10-p147.pdf">Fibonacci numbers, ordered partitions, and transformations of a finite set</a>, Australasian Journal of Combinatorics, Volume 10(1994), pp. 147-151. See triangle p. 151 (with rows reversed and initial term 0).
%F A193923 T(n, k) = Sum_{p=0..k} binomial(n+k-p-1, p). - _Johannes W. Meijer_, Aug 12 2013
%F A193923 T(n, n) = Fibonacci(2*n) for n>=1. - _Michel Marcus_, Nov 03 2020
%e A193923 First six rows:
%e A193923 1
%e A193923 1...1
%e A193923 1...2...3
%e A193923 1...3...5....8
%e A193923 1...4...8....13...21
%e A193923 1...5...12...21...34...55
%p A193923 T := proc(n, k) option remember: if k = 0 then return(1) fi: if k = n then return(combinat[fibonacci](2*n)) fi: T(n, k) := T(n-1, k-1) + T(n-1, k) end: seq(seq(T(n, k), k=0..n), n=0..9); # _Johannes W. Meijer_, Aug 12 2013
%t A193923 p[n_, x_] := (x + 1)^n;
%t A193923 q[n_, x_] := Sum[Fibonacci[k + 1]*x^(n - k), {k, 0, n}];
%t A193923 t[n_, k_] := Coefficient[p[n, x], x^k]; t[n_, 0] := p[n, x] /. x -> 0;
%t A193923 w[n_, x_] := Sum[t[n, k]*q[n + 1 - k, x], {k, 0, n}]; w[-1, x_] := 1
%t A193923 g[n_] := CoefficientList[w[n, x], {x}]
%t A193923 TableForm[Table[Reverse[g[n]], {n, -1, z}]]
%t A193923 Flatten[Table[Reverse[g[n]], {n, -1, z}]]  (* A193923 *)
%t A193923 TableForm[Table[g[n], {n, -1, z}]]
%t A193923 Flatten[Table[g[n], {n, -1, z}]]  (* A193924 *)
%Y A193923 Cf. A193722, A193924.
%Y A193923 Cf. A001906 (Fibonacci(2*n)).
%K A193923 nonn,tabl
%O A193923 0,5
%A A193923 _Clark Kimberling_, Aug 09 2011