

A102426


Triangle read by rows giving coefficients of polynomials defined by F(0,x)=0, F(1,x)=1, F(n,x) = x*F(n1,x) + F(n2,x).


20



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, 15, 1
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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 p. 19).  Tom Copeland, Oct 11 2014
Aside from the initial zeros, these are the antidiagonals read from bottom to top of the numerical coefficients of the MaurerCartan form matrix of the Leibniz group L^(n)(1,1) presented on p. 9 of the Olver paper, which is generated as exp[c. * M] with (c.)^n = c_n and M the Lie infinitesimal generator A218272. Reverse of A011973.  Tom Copeland, Jul 02 2018


REFERENCES

Dominique Foata and GuoNiu Han, Multivariable tangent and secant qderivative polynomials, Manuscript, Mar 21 2012


LINKS

G. C. Greubel, Rows n = 0..100 of triangle, flattened
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.
Dominique Foata and GuoNiu Han, Multivariable tangent and secant qderivative polynomials, Moscow Journal of Combinatorics and Number Theory, vol. 2, issue 3, 2012, pp. 3484, [pp. 232282].
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
P. Olver, The canonical contact form.
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! = lim_{c>0} 1/(k+c)!.  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..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
2*x + 1
x^2 + 3*x + 1
3*x^2 + 4*x + 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

Join[{0}, Table[ Select[ CoefficientList[ Fibonacci[n, x], x], 0 < # &], {n, 0, 17}]//Flatten] (* Clark Kimberling, Oct 10 2013 and slightly modified by Robert G. Wilson v, May 03 2017 *)


PROG

(MAGMA) [0] cat [Binomial(Floor(n/2)+k, Floor((n1)/2k) ): k in [0..Floor((n1)/2)], n in [0..17]]; // G. C. Greubel, Oct 13 2019


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 A320250 A089141
Adjacent sequences: A102423 A102424 A102425 * A102427 A102428 A102429


KEYWORD

easy,nonn,tabf


AUTHOR

Russell Walsmith, Jan 08 2005


STATUS

approved



