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A094440 Triangular array T(n,k) = F(n+1-k)*C(n,k-1), k = 1,2,3,...,n; n >= 1. 6
1, 1, 2, 2, 3, 3, 3, 8, 6, 4, 5, 15, 20, 10, 5, 8, 30, 45, 40, 15, 6, 13, 56, 105, 105, 70, 21, 7, 21, 104, 224, 280, 210, 112, 28, 8, 34, 189, 468, 672, 630, 378, 168, 36, 9, 55, 340, 945, 1560, 1680, 1260, 630, 240, 45, 10, 89, 605, 1870, 3465, 4290, 3696, 2310, 990, 330 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

1,3

COMMENTS

Row sums yield the even-subscripted Fibonacci numbers (A001906).

LINKS

Table of n, a(n) for n=1..64.

FORMULA

From Peter Bala, Aug 17 2007: (Start)

With an offset of 0, the row polynomials F(n,x) = Sum_{k = 0..n} C(n,k)*Fibonacci(n-k)*x^k satisfy F(n,x)*L(n,x) = F(2*n,x), where L(n,x) = Sum_{k = 0..n} C(n,k)*Lucas(n-k)*x^k.

Other identities and formulas include:

F(n+1,x)^2 - F(n,x)*F(n+2,x) = (x^2 + x - 1)^n;

Sum_{k = 0..n} C(n,k)*F(n-k,x)*L(k,x) = 2^n F(n,x);

F(n,2*x) = Sum_{k = 0..n} C(n,k)*F(n-k,x)*x^k;

F(n,3*x) = Sum_{k = 0..n} C(n,k)*F(n-k,2*x)*x^k, etc.

Sequence {F(n,r)}n>=1 gives the r th binomial transform of the Fibonacci numbers: r = 1 gives A001906, r = 2 gives A030191, r = 3 gives A099453, r = 4 gives A081574, r = 5 gives A081574.

F(n,1/phi) = (-1)^(n-1)*F(n,-phi) = sqrt(5)^(n-1) for n >= 1, where phi = (1 + sqrt(5))/2.

The polynomials F(n,-x) satisfy a Riemann hypothesis: the zeros of F(n,-x) lie on the vertical line Re x = 1/2 in the complex plane.

G.f.: t/(1 - (2*x + 1)*t + (x^2 + x - 1)*t^2) = t + (1 + 2*x)*t^2 + (2 + 3*x + 3*x^2)*t^3 + (3 + 8*x + 6*x^2 + 4*x^3)*t^4 + ... . (End)

From Peter Bala, Jun 29 2016: (Start)

Working with an offset of 0, the n-th row polynomial F(n,x) = 1/sqrt(5)*( (x + phi)^n - (x - 1/phi)^n ), where phi = (1 + sqrt(5))/2.

d/dx(F(n,x)) = n*F(n-1,x).

F(-n,x) = -F(n,x)/(x^2 + x - 1)^n.

F(n,x - 1) = (-1)^(n-1)*F(n,-x).

F(n,x) is a divisibility sequence of polynomials, that is, if n divides m then F(n,x) divides F(m,x) in the polynomial ring Z[x]. (End)

EXAMPLE

Triangle starts:

1;

1, 2;

2, 3, 3;

3, 8, 6, 4;

...

T(4,3) = F(2)*C(4,2) = 1*6 = 6.

MAPLE

with(combinat): T:=(n, k)->binomial(n, k-1)*fibonacci(n+1-k): for n from 1 to 11 do seq(T(n, k), k=1..n) od; # yields sequence in triangular form # Emeric Deutsch

MATHEMATICA

Table[Fibonacci[n+1-k]Binomial[n, k-1], {n, 20}, {k, n}]//Flatten (* Harvey P. Dale, Sep 14 2016 *)

PROG

(MAGMA) /* As triangle */ [[Fibonacci(n+1-k)*Binomial(n, k-1): k in [1..n]]: n in [1.. 15]]; // Vincenzo Librandi, Aug 15 2017

CROSSREFS

Cf. A000045, A001906, A030191, A081574, A094435, A132148.

Sequence in context: A025496 A099959 A099964 * A093736 A257481 A014589

Adjacent sequences:  A094437 A094438 A094439 * A094441 A094442 A094443

KEYWORD

nonn,tabl

AUTHOR

Clark Kimberling, May 03 2004

EXTENSIONS

Error in expansion of generating function corrected by Peter Bala, Sep 24 2008

STATUS

approved

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Last modified June 25 17:47 EDT 2019. Contains 324353 sequences. (Running on oeis4.)