A193923 - OEIS

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

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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...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.