login
This site is supported by donations to The OEIS Foundation.

 

Logo

Annual Appeal: Today, Nov 11 2014, is the 4th anniversary of the launch of the new OEIS web site. 70,000 sequences have been added in these four years, all edited by volunteers. Please make a donation (tax deductible in the US) to help keep the OEIS running.

Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A102426 Triangle read by rows giving coefficients of polynomials defined by F(0)=0, F(1)=1, F(n+1) = F(n) + x*F(n-1). 14
0, 1, 1, 1, 1, 2, 1, 1, 3, 1, 3, 4, 1, 1, 6, 5, 1, 4, 10, 6, 1, 1, 10, 15, 7, 1, 5, 20, 21, 8, 1, 1, 15, 35, 28, 9, 1, 6, 35, 56, 36, 10, 1, 1, 21, 70, 84, 45, 11, 1, 7, 56, 126, 120, 55, 12, 1, 1, 28, 126, 210, 165, 66, 13, 1, 8, 84, 252, 330, 220, 78, 14, 1, 1, 36, 210, 462, 495, 286, 91 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,6

COMMENTS

Essentially the same as A098925: a(0)=0 followed by A098925. [R. J. Mathar, Aug 30 2008]

F(n) + 2x * F(n-1) gives Lucas polynomials (cf. A034807). - Maxim Krikun (krikun(AT)iecn.u-nancy.fr), Jun 24 2007

After the initial 0, these are the nonzero coefficients of the Fibonacci polynomials; see the Mathematica section. - Clark Kimberling, Oct 10 2013

Aside from signs and index shift, the coefficients of the characteristic polynomial of the Coxeter adjacency matrix for the Coxeter group A_n related to the Chebyshev polynomial of the second kind (cf. Damianou link pg. 19). - Tom Copeland, Oct 11 2014

REFERENCES

Dominique Foata and Guo-Niu Han, Multivariable tangent and secant q-derivative polynomials, Manuscript, Mar 21 2012

LINKS

Table of n, a(n) for n=0..79.

H.-H. Chern, H.-K. Hwang, T.-H. Tsai, Random unfriendly seating arrangement in a dining table, arXiv preprint arXiv:1406.0614, 2014

P. Damianou, On the characteristic polynomials of Cartan matrices and Chebyshev polynomials, arXiv preprint arXiv:1110.6620, 2014.

G. Hetyei, Hurwitzian continued fractions containing a repeated constant and an arithmetic progression, arXiv preprint arXiv:1211.2494, 2012. - From N. J. A. Sloane, Jan 02 2013

FORMULA

Alternatively, as n is even or odd: T(n-2, k) + T(n-1, k-1) = T(n, k), T(n-2, k) + T(n-1, k) = T(n, k)

T(n, k)=binomial(floor(n/2)+k, floor((n-1)/2-k) ) - Paul Barry, Jun 22 2005

Beginning with the second polynomial in the example and offset=0, P(n,t)= sum(j=0,..,n, binom(n-j,j)*x^j) with the convention that 1/k! is zero for k=-1,-2,..., i.e., 1/k!=limit 1/(k+a)! as a tends to zero. - Tom Copeland, Oct 11 2014

EXAMPLE

The first few polynomials are:

0

1

1

x + 1

2x + 1

x^2 + 3x + 1

3x^2 + 4x + 1

MATHEMATICA

Table[Fibonacci[n, x], {n, 0, 10}] (* Clark Kimberling, Oct 10 2013 *)

CROSSREFS

Upward diagonals sums are A062200. Downward rows are A102427. Row sums are A000045. Row terms reversed = A011973. Also A102427, A102428, A102429.

All of A011973, A092865, A098925, A102426, A169803 describe essentially the same triangle in different ways.

Sequence in context: A035667 A092865 A098925 * A052920 A089141 A245717

Adjacent sequences:  A102423 A102424 A102425 * A102427 A102428 A102429

KEYWORD

easy,nonn,tabf

AUTHOR

Russell Walsmith, Jan 08 2005

STATUS

approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Transforms | Superseeker | Recent | More pages
The OEIS Community | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement .

Last modified December 22 12:59 EST 2014. Contains 252357 sequences.