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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).
2
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
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. A001906 (Fibonacci(2*n)).
Sequence in context: A194740 A194762 A054250 * A198811 A067337 A180091
KEYWORD
nonn,tabl
AUTHOR
Clark Kimberling, Aug 09 2011
STATUS
approved