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Array of coefficients of polynomials related to integer powers of the generating function of Catalan numbers A000108.
22

%I #138 Mar 21 2024 20:57:12

%S 1,1,1,-1,1,-2,1,-3,1,1,-4,3,1,-5,6,-1,1,-6,10,-4,1,-7,15,-10,1,1,-8,

%T 21,-20,5,1,-9,28,-35,15,-1,1,-10,36,-56,35,-6,1,-11,45,-84,70,-21,1,

%U 1,-12,55,-120,126,-56,7,1,-13,66,-165,210,-126,28,-1,1,-14,78,-220,330,-252,84,-8,1,-15,91,-286,495,-462,210,-36,1

%N Array of coefficients of polynomials related to integer powers of the generating function of Catalan numbers A000108.

%C This is a signed version of A011973 (Fibonacci polynomials) with different offset.

%C The sequence of row lengths is [1,1,2,2,3,3,4,4,5,5,6,6,...] = A008619(n-1), n>=1.

%C The row sums give the period 6 sequence [1,1,0,-1,-1,0,...] = A010892(n-1), n>=1.

%C The o.g.f. for the column m sequence (with leading zeros) is ((-1)^m)*x^(2*m+1)/(1-x)^(m+1).

%C The unsigned row sums give the Fibonacci numbers A000045(n-1), n>=1.

%C The row polynomial are P(n,x):= Sum_{m=0..ceiling(n/2)-1} a(n,m)*x^m = (sqrt(x)^(n-1))*S(n-1,1/sqrt(x)), n>=1, with Chebyshev's S(n,x) polynomials A049310.

%C These polynomials appear in the formula 1/c(x)^n = P(n+1,x) - x*P(n,x)*c(x), n>=1, with the o.g.f. c(x):=(1-sqrt(1-4*x))/(2*x) of A000108 (Catalan numbers). See the W. Lang reference, eqs. (1) and (2), p. 408, with P(n,x):=p(-n,x).

%C These polynomials also appear in the formula c(x)^n = (-P(n-1,x) + P(n,x)*c(x))/x^(n-1), n>=1, with the above given o.g.f. c(x) of A000108 (Catalan numbers). See the W. Lang reference, eq. (1), with P(n,x):=p(-n,x).

%C With offset n>=0 this array a(n,m) coincides with the row reversed coefficient table of Chebyshev's S-polynomials without interspersed zeros. See A049310 for the S(n,x) coefficient table with increasing powers of x.

%C The polynomials with this sequence as coefficients form the set of so-called "Catalan polynomials", having arisen from computations in looking at the problem of 'fitting' iterated generating function schemes to the Catalan sequence. A neighboring pair forms the basis of a first-order linear recurrence that generates, through a succession of iterated generating functions (polynomials in Z[x]), a predetermined number of Catalan numbers before 'failing' - see the Clapperton et al. 2008 reference in Utilitas Mathematica, where some of the essential mathematical properties of the Catalan polynomials are also listed (based mainly on existing results for Dickson and Chebyshev polynomials, to which they are related). - _Peter J Larcombe_, Sep 16 2008

%C In the Clapperton et al. 2008 Congressus Numerantium paper, a new class of nonlinear identities satisfied by Catalan polynomials are presented. They arise from the algebraic implementation of particular cases of a general root finding formulation due to Householder, of which the classic O(2) Newton-Raphson and O(3) Halley algorithms are special cases. The role of Catalan polynomials in forming Padé approximants to the Catalan sequence o.g.f. is also discussed. - _Peter J Larcombe_, Nov 02 2008

%C These polynomials appear in the following statements: (i) P(k+1,x)/P(k+2,x) is the g.f. of all ordered trees (Dyck paths) of height at most k; (ii) x^k/(P(k+1,x)*P(k+2,x)) is the g.f. of all ordered trees (Dyck paths) of height k. See the de Bruijn et al., the Kreweras, the Sedgewick and Flajolet (p. 258), and the Flajolet and Sedgewick (p. 326) references. - _Emeric Deutsch_, Jun 16 2011

%C For a mirrored, shifted version showing the relation of these coefficients to the Pascal triangle, Fibonacci, and other number triangles, see A030528. See also A053122 for a relation to Cartan matrices. (Cf. A011973, A169803, A115139, A092865, A098925, and A102426.) - _Tom Copeland_, Nov 04 2014

%C _M. Sinan Kul_, Dec 09 2015, observed that (in a rewritten form) Chebyshev's S polynomials A049310

%C are given by S(n, x) = Sum_{m=0..floor(n/2)} a(n+1, m)*x^(n-2*m), n >= 0. This formula is well known and can be proved from the S recurrence by induction using the recurrence for the binomial coefficients. - _Wolfdieter Lang_, Feb 01 2016

%C These are the coefficients of generalized Fibonacci polynomials (see link bellow). - _Rigoberto Florez_, Aug 28 2022

%D J. A. Clapperton, P. J. Larcombe and E. J. Fennessey, On iterated generating functions for integer sequences and Catalan polynomials, Utilitas Mathematica, 77 (2008), 3-33.

%D J. A. Clapperton, P. J. Larcombe, E. J. Fennessey and P. Levrie, A class of auto-identities for Catalan polynomials and Padé approximation, Congressus Numerantium, 189 (2008), 77-95.

%D N. G. de Bruijn, D. E. Knuth and S. O. Rice, The average height of planted plane trees, in: Graph Theory and Computing (ed. T. C. Read), Academic Press, New York, 1972, pp. 15-22.

%D R. Sedgewick and P. Flajolet, An Introduction to the Analysis of Algorithms, Addison-Wesley, Reading, MA, 1996.

%H Alois P. Heinz, <a href="/A115139/b115139.txt">Rows n = 1..200, flattened</a>

%H Giulio Cerbai, Anders Claesson, and Luca Ferrari, <a href="https://arxiv.org/abs/1907.08142">Stack sorting with restricted stacks</a>, arXiv:1907.08142 [cs.DS], 2019.

%H Philippe Flajolet and Robert Sedgewick, <a href="http://algo.inria.fr/flajolet/Publications/AnaCombi/anacombi.html">Analytic Combinatorics</a>, Cambridge Univ. Press, 2009.

%H R. Florez and J. C. Saunders, <a href="http://math.colgate.edu/~integers/w69/w69.pdf">Irreducibility of generalized Fibonacci polynomials </a>, Integers 22 (2022).

%H Atsushi Komaba, Hisashi Johno, and Kazunori Nakamoto, <a href="https://arxiv.org/abs/2206.03166">A novel statistical approach for two-sample testing based on the overlap coefficient</a>, arXiv:2206.03166 [math.ST], 2022.

%H G. Kreweras, <a href="http://www.numdam.org/item?id=BURO_1970__15__3_0">Sur les éventails de segments</a>, Cahiers du Bureau Universitaire de Recherche Opérationelle, Cahier no. 15, Paris, 1970, pp. 3-41.

%H Wolfdieter Lang, <a href="/A115139/a115139.pdf">First 16 rows</a>

%H Wolfdieter Lang, <a href="http://www.fq.math.ca/Scanned/38-5/lang.pdf">On polynomials related to powers of the generating function of Catalan's numbers</a>, Fib. Quart. 38,5 (2000) 408-419.

%H Peter J. Larcombe and Eric J. Fennessey, <a href="https://www.fq.math.ca/Papers1/52-1/LarcombeFennessey.pdf">A Non-Linear Identity for A Particular Class of Polynomial Families</a>, Fibonacci Quart. 52 (2014), no. 1, 75-79. Mentions this sequence.

%F a(n, m) = ((-1)^(m))*binomial(n-1-m, m), n>=1, m=0..ceiling(n/2)-1.

%F a(n,m) = [x^m]P(n,x), n>=1, m=0..ceiling(n/2)-1, with P(n,x) given above in terms of Chebyshev's S-polynomials.

%F P(n,x) = (u^(2*n) - v^(2*n))/(u^2 - v^2), where u and v are defined by u^2 + v^2 =1 and u*v = sqrt(x). Example: P(3,x) = (u^6 - v^6)/(u^2 - v^2) = u^4 + u^2*v^2 + v^4 = 1 - x. - _Emeric Deutsch_, Jun 16 2011

%F G.f.: 1/(1- x + y*x^2) = R(0)/2, where R(k) = 1 + 1/(1 - (2*k+1- x*y)*x/((2*k+2- x*y)*x + 1/R(k+1) )); (continued fraction). - _Sergei N. Gladkovskii_, Nov 09 2013

%F T(n, k) = GegenbauerC(k, (n+1)/2-k, -1)) assuming the triangle (0,0) based. - _Peter Luschny_, May 10 2016

%e The irregular triangle a(n, m) begins:

%e n\m 0 1 2 3 4 5 6 7 8 9

%e 1: 1

%e 2: 1

%e 3: 1 -1

%e 4: 1 -2

%e 5: 1 -3 1

%e 6: 1 -4 3

%e 7: 1 -5 6 -1

%e 8: 1 -6 10 -4

%e 9: 1 -7 15 -10 1

%e 10: 1 -8 21 -20 5

%e 11: 1 -9 28 -35 15 -1

%e 12: 1 -10 36 -56 35 -6

%e 13: 1 -11 45 -84 70 -21 1

%e 14: 1 -12 55 -120 126 -56 7

%e 15: 1 -13 66 -165 210 -126 28 -1

%e 16: 1 -14 78 -220 330 -252 84 -8

%e 17: 1 -15 91 -286 495 -462 210 -36 1

%e 18: 1 -16 105 -364 715 -792 462 -120 9

%e 19: 1 -17 120 -455 1001 -1287 924 -330 45 -1

%e 20: 1 -18 136 -560 1365 -2002 1716 -792 165 -10

%e ... Reformatted and extended. - _Wolfdieter Lang_, Jan 27 2016

%e 1/c(x) = P(2,x) - x*P(1,x)*c(x) = 1 - x*c(x), with the o.g.f. of A000108 (Catalan).

%e 1/c(x)^2 = P(3,x) - x*P(2,x)*c(x) = (1-x) - x*c(x).

%e c(x)^2 = (-P(1,x) + P(2,x)*c(x))/x^1 = (-1 + 1*c(x))/x.

%e c(x)^3 = (-P(2,x) + P(3,x)*c(x))/x^2 = (-1 + (1-x)*c(x))/x^2.

%e P(3,x) = 1-x = x*S(2,1/sqrt(x))) with Chebyshev's S(2,y) = U(2,y/2) = y^2 - 1.

%p seq(seq((-1)^k*binomial(n-k,k),k=0..floor(n/2)),n=0..16); # _Peter Luschny_, May 10 2016

%t p[x_, n_] := p[x, n] = p[x, n - 1] + x*p[x, n - 2];

%t p[x_, -1] = p[x_, 0] = 1; p[x_, 1] = 1 + x;

%t Flatten[ Table[ CoefficientList[p[-x, n - 1], x], {n, 0, 16}]]

%t (* _Jean-François Alcover_, Jun 20 2011 *)

%t Flatten[Map[CoefficientList[#,x]&, Table[Sum[Binomial[t - i, i] x^(i) (-1)^i, {i, 0, t}], {t, 1,15}]]] (* _Rigoberto Florez_, Aug 28 2022 *)

%o (Python)

%o import math

%o L1 = [math.comb(t - i, i)*(-1)**i for t in range(16) for i in range(t)]

%o L1 = list(filter((0).__ne__, L1))

%o print(L1) # _Rigoberto Florez_, Sep 03 2022

%Y Cf. A011973, A030528, A053122, A092865, A098925, A102426, A115139, A169803.

%K sign,easy,tabf

%O 1,6

%A _Wolfdieter Lang_, Jan 13 2006