A102426 Triangle read by rows giving coefficients of polynomials defined by F(0,x)=0, F(1,x)=1, F(n,x) = F(n-1,x) + x*F(n-2,x).

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

Offset: 0

Author

Russell Walsmith, Jan 08 2005

Keywords

• easy
• nonn
• tabf

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 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 Maurer-Cartan 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

Examples

			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)
		

References

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

Links

• G. C. Greubel, Rows n = 0..100 of triangle, flattened
• R. Andre-Jeannin, A generalization of Morgan-Voyce polynomials, The Fibonacci Quarterly 32.3 (1994): 228-31.
• H.-H. Chern, H.-K. Hwang, and 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): 149-55.
• Dominique Foata and Guo-Niu Han, Multivariable tangent and secant q-derivative polynomials, Moscow Journal of Combinatorics and Number Theory, vol. 2, issue 3, 2012, pp. 34-84, [pp. 232-282].
• 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, 2005.
• Z. Trzaska, On Fibonacci hyberbolic geometry and modified number triangles, Fibonacci Quarterly, 34.2 (1996): 129-38.

Crossrefs

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

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

Cf. A078812, A085478.

Programs

Magma

[0] cat [Binomial(Floor(n/2)+k, Floor((n-1)/2-k) ): k in [0..Floor((n-1)/2)], n in [0..17]]; // G. C. Greubel, Oct 13 2019

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 *)

PARI

F(n) = if (n==0, 0, if (n==1, 1, F(n-1) + x*F(n-2)));
tabf(nn) = for (n=0, nn, print(Vec(F(n)))); \\ Michel Marcus, Feb 10 2020

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}, binomial(n-j,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*(1-x^2) (odd)) / (1 - (2+t)*x^2 + x^4).

Recursion relations:

A) p(n,t) = p(n-1,t) + p(n-2,t) for n=2,4,6,8,...

B) p(n,t) = t*p(n-1,t) + p(n-2,t) for n=3,5,7,...

C) a(n,k) = a(n-2,k) + a(n-1,k) for n=4,6,8,...

D) a(n,k) = a(n-2,k) + a(n-1,k-1) 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 Andre-Jeannine for the generalized Morgan-Voyce 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(n-4,t;1) for n=4,6,8,... .

The even and odd polynomials are also presented in Trzaska and Ferri.

Dropping the initial 0 and re-indexing 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(n-k,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 (1-t)

IFb(4,t) = 2 (1-t)

IFb(5,t) = -3 (1-t)

IFb(6,t) = 5 (1-t)

IFb(7,t) = -8 (1-t)

IFb(8,t) = 13 (1-t)

... .

(End)

Extensions

Name corrected by John K. Sikora, Feb 10 2020