A102426 - OEIS

%I #102 Sep 08 2022 08:45:16

%S 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,

%T 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,

%U 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

%N 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).

%C Essentially the same as A098925: a(0)=0 followed by A098925. - _R. J. Mathar_, Aug 30 2008

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

%C After the initial 0, these are the nonzero coefficients of the Fibonacci polynomials; see the Mathematica section. - _Clark Kimberling_, Oct 10 2013

%C 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

%C 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

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

%H G. C. Greubel, <a href="/A102426/b102426.txt">Rows n = 0..100 of triangle, flattened</a>

%H R. Andre-Jeannin, <a href="http://www.fq.math.ca/Scanned/32-3/andre-jeannin.pdf">A generalization of Morgan-Voyce polynomials</a>, The Fibonacci Quarterly 32.3 (1994): 228-31.

%H H.-H. Chern, H.-K. Hwang, and T.-H. Tsai, <a href="http://arxiv.org/abs/1406.0614">Random unfriendly seating arrangement in a dining table</a>, arXiv preprint arXiv:1406.0614 [math.PR], 2014.

%H T. Copeland, <a href="http://tcjpn.wordpress.com/2015/10/12/the-elliptic-lie-triad-kdv-and-ricattt-equations-infinigens-and-elliptic-genera/">Addendum to Elliptic Lie Triad</a>

%H P. Damianou, <a href="http://arxiv.org/abs/1110.6620">On the characteristic polynomials of Cartan matrices and Chebyshev polynomials</a>, arXiv preprint arXiv:1110.6620 [math.RT], 2014.

%H G. Ferri, <a href="http://www.fq.math.ca/Scanned/35-2/ferri.pdf">The appearance of Fibonacci and Lucas numbers in the simulation of electrical power lines supplied by two sides</a>, The Fibonacci Quarterly 35.2 (1997): 149-55.

%H Dominique Foata and Guo-Niu Han, <a href="http://irma.math.unistra.fr/~foata/paper/pub119.html">Multivariable tangent and secant q-derivative polynomials</a>, Moscow Journal of Combinatorics and Number Theory, vol. 2, issue 3, 2012, pp. 34-84, [pp. 232-282].

%H G. Hetyei, <a href="http://arxiv.org/abs/1211.2494">Hurwitzian continued fractions containing a repeated constant and an arithmetic progression</a>, arXiv preprint arXiv:1211.2494 [math.CO], 2012. - From _N. J. A. Sloane_, Jan 02 2013

%H P. Olver, <a href="http://www-users.math.umn.edu/~olver/di_/contact.pdf">The canonical contact form</a>, 2005.

%H Z. Trzaska, <a href="http://www.fq.math.ca/Scanned/34-2/trzaska.pdf">On Fibonacci hyberbolic geometry and modified number triangles</a>, Fibonacci Quarterly, 34.2 (1996): 129-38.

%F 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)

%F T(n, k) = binomial(floor(n/2)+k, floor((n-1)/2-k) ). - _Paul Barry_, Jun 22 2005

%F 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

%F From _Tom Copeland_, Jan 19 2016: (Start)

%F 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).

%F Recursion relations:

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

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

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

%F D) a(n,k) = a(n-2,k) + a(n-1,k-1) for n=3,5,7,...

%F 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,... .

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

%F 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.

%F 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:

%F IFb(0,t) =  1

%F IFb(1,t) = -1

%F IFb(2,t) = -t

%F IFb(3,t) = -1 (1-t)

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

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

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

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

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

%F ... .

%F (End)

%e The first few polynomials are:

%e   0

%e   1

%e   1

%e   x + 1

%e   2*x + 1

%e   x^2 + 3*x + 1

%e   3*x^2 + 4*x + 1

%e ------------------

%e From _Tom Copeland_, Jan 19 2016: (Start)

%e [n]:

%e 0:  0

%e 1:  1

%e 2:  1

%e 3:  1  1

%e 4:  2  1

%e 5:  1  3  1

%e 6:  3  4  1

%e 7:  1  6  5   1

%e 8:  4 10  6   1

%e 9:  1 10 15   7   1

%e 10: 5 20 21   8   1

%e 11: 1 15 35  28   9  1

%e 12: 6 35 56  36  10  1

%e 13: 1 21 70  84  45 11 1

%e (End)

%t 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 *)

%o (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

%o (PARI) F(n) = if (n==0, 0, if (n==1, 1, F(n-1) + x*F(n-2)));

%o tabf(nn) = for (n=0, nn, print(Vec(F(n)))); \\ _Michel Marcus_, Feb 10 2020

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

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

%Y Cf. A078812, A085478.

%K easy,nonn,tabf

%O 0,6

%A _Russell Walsmith_, Jan 08 2005

%E Name corrected by _John K. Sikora_, Feb 10 2020