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
Examples
First six rows: 1 1...1 1...2...3 1...3...5....8 1...4...8....13...21 1...5...12...21...34...55
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).
Programs
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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
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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 *)
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
Comments