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).

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, 1, 7, 23, 50, 88, 144, 233, 377, 1, 8, 30, 73, 138, 232, 377, 610, 987, 1, 9, 38, 103, 211, 370, 609, 987, 1597, 2584, 1, 10, 47, 141, 314, 581, 979, 1596, 2584, 4181, 6765

Offset

0,5

Comments

See A193722 for the definition of fusion of two sequences of polynomials or triangular arrays.

The row sums equal A079289(2*n). - Johannes W. Meijer, Aug 12 2013

Links

Michel Marcus, Rows n=0..100 of triangle, flattened

T. G. Lavers, Fibonacci numbers, ordered partitions, and transformations of a finite set, Australasian Journal of Combinatorics, Volume 10(1994), pp. 147-151. See triangle p. 151 (with rows reversed and initial term 0).

Formula

T(n, k) = Sum_{p=0..k} binomial(n+k-p-1, p). - Johannes W. Meijer, Aug 12 2013

T(n, n) = Fibonacci(2*n) for n>=1. - Michel Marcus, Nov 03 2020

Example

First six rows:
1
1...1
1...2...3
1...3...5....8
1...4...8....13...21
1...5...12...21...34...55

Maple

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

Mathematica

p[n_, x_] := (x + 1)^n;
q[n_, x_] := Sum[Fibonacci[k + 1]*x^(n - k), {k, 0, n}];
t[n_, k_] := Coefficient[p[n, x], x^k]; t[n_, 0] := p[n, x] /. x -> 0;
w[n_, x_] := Sum[t[n, k]*q[n + 1 - k, x], {k, 0, n}]; w[-1, x_] := 1
g[n_] := CoefficientList[w[n, x], {x}]
TableForm[Table[Reverse[g[n]], {n, -1, z}]]
Flatten[Table[Reverse[g[n]], {n, -1, z}]]  (* A193923 *)
TableForm[Table[g[n], {n, -1, z}]]
Flatten[Table[g[n], {n, -1, z}]]  (* A193924 *)

Crossrefs

Cf. A193722, A193924.

Cf. A001906 (Fibonacci(2*n)).

Sequence in context: A194740 A194762 A054250 * A198811 A067337 A180091

Adjacent sequences: A193920 A193921 A193922 * A193924 A193925 A193926

Keyword

nonn,tabl

Author

Clark Kimberling, Aug 09 2011

Status

approved