

A102426


Triangle read by rows giving coefficients of polynomials defined by F(0)=0, F(1)=1, F(n+1) = F(n) + x*F(n1).


18



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
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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(n1) gives Lucas polynomials (cf. A034807).  Maxim Krikun (krikun(AT)iecn.unancy.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 GuoNiu Han, Multivariable tangent and secant qderivative polynomials, Manuscript, Mar 21 2012


LINKS

Table of n, a(n) for n=0..79.
R. AndreJeannin, A generalization of MorganVoyce polynomials, The Fibonacci Quarterly 32.3 (1994): 22831.
H.H. Chern, H.K. Hwang, T.H. Tsai, Random unfriendly seating arrangement in a dining table, arXiv preprint arXiv:1406.0614 [math.PR], 2014.
T. Copeland, Addendum to Elliptic Lie Triad
P. Damianou, On the characteristic polynomials of Cartan matrices and Chebyshev polynomials, arXiv preprint arXiv:1110.6620 [math.RT], 2014.
G. Ferri, The appearance of Fibonacci and Lucas numbers in the simulation of electrical power lines supplied by two sides, The Fibonacci Quarterly 35.2 (1997): 14955.
G. Hetyei, Hurwitzian continued fractions containing a repeated constant and an arithmetic progression, arXiv preprint arXiv:1211.2494 [math.CO], 2012.  From N. J. A. Sloane, Jan 02 2013
Z. Trzaska, On Fibonacci hyberbolic geometry and modified number triangles, Fibonacci Quarterly, 34.2 (1996): 12938.


FORMULA

Alternatively, as n is even or odd: T(n2, k) + T(n1, k1) = T(n, k), T(n2, k) + T(n1, k) = T(n, k)
T(n, k)=binomial(floor(n/2)+k, floor((n1)/2k) )  Paul Barry, Jun 22 2005
Beginning with the second polynomial in the example and offset=0, P(n,t)= sum(j=0,..,n, binomial(nj,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
From Tom Copeland, Jan 19 2016: (Start)
O.g.f.: [x + x^2  x^3] / [1  (2+t)x^2 + x^4] = [x^2 (even part) + x(1x^2) (odd)] / [1  (2+t)x^2 + x^4].
Recursion relations:
A) p(n,t) = p(n1,t) + p(n2,t) for n=2,4,6,8,...
B) p(n,t) = t p(n1,t) + p(n2,t) for n=3,5,7,...
C) a(n,k) = a(n2,k) + a(n1,k) for n=4,6,8,...
D) a(n,k) = a(n2,k) + a(n1,k1) for n=3,5,7,...
Relation A generalized to MV(n,t;r) = P(2n+1,t) + r R(2n,t) for n=1,2,3,.. (cf. A078812 and A085478) is the generating relation on p. 229 of AndreJeannine for the generalized MorganVoyce polynomials, e.g., MV(2,t;r) = p(5,t) + r p(4,t) = (1 + 3t + t^2) + r (2 + t) = (1 + 2r) + (3 + r) t + t^2, so P(n,t) = MV(n4,t;1) for n=4,6,8,.. .
The even and odd polynomials are also presented in Trzaska and Ferri.
Dropping the initial 0 and reindexing with initial m=0 gives the row polynomials Fb(m,t) = p(n+1,t) below with o.g.f. G(t,x)/x, starting with Fb(0,t) = 1, Fb(1,t) = 1, Fb(2,t) = 1 + t, and Fb(3,t) = 2 + t .
The o.g.f. x/G(x,t) = [1  (2+t)x^2 + x^4] / [1 + x  x^2] then generates a sequence of polynomials IFb(t) such that the convolution Sum(k=0 to n) IFb(nk,t) Fb(k,t) vanishes for n>1 and is one for n=0. These linear polynomials have the basic Fibonacci numbers A000045 as an overall factor:
IFb(0,t) = 1
IFb(1,t) = 1,
IFb(2,t) = t
IFb(3,t) = 1 (1t)
IFb(4,t) = 2 (1t)
IFb(5,t) = 3 (1t)
IFb(6,t) = 5 (1t)
IFb(7,t) = 8 (1t)
IFb(8,t) = 13 (1t)
... .
(End)


EXAMPLE

The first few polynomials are:
0
1
1
x + 1
2x + 1
x^2 + 3x + 1
3x^2 + 4x + 1

From Tom Copeland, Jan 19 2016: (Start)
[n]:
0: 0
1: 1
2: 1
3: 1 1
4: 2 1
5: 1 3 1
6: 3 4 1
7: 1 6 5 1
8: 4 10 6 1
9: 1 10 15 7 1
10: 5 20 21 8 1
11: 1 15 35 28 9 1
12: 6 35 56 36 10 1
13: 1 21 70 84 45 11 1
(End)


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.
Cf. A078812, A085478.
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



