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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). 10
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

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

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

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

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Last modified September 22 06:13 EDT 2014. Contains 247039 sequences.