This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A034807 #238 Apr 24 2025 08:54:17
%S A034807 2,1,1,2,1,3,1,4,2,1,5,5,1,6,9,2,1,7,14,7,1,8,20,16,2,1,9,27,30,9,1,
%T A034807 10,35,50,25,2,1,11,44,77,55,11,1,12,54,112,105,36,2,1,13,65,156,182,
%U A034807 91,13,1,14,77,210,294,196,49,2,1,15,90,275,450,378,140,15,1,16,104
%N A034807 Triangle T(n,k) of coefficients of Lucas (or Cardan) polynomials.
%C A034807 These polynomials arise in the following setup. Suppose G and H are power series satisfying G + H = G*H = 1/x. Then G^n + H^n = (1/x^n)*L_n(-x).
%C A034807 Apart from signs, triangle of coefficients when 2*cos(nt) is expanded in terms of x = 2*cos(t). For example, 2*cos(2t) = x^2 - 2, 2*cos(3t) = x^3 - 3x and 2*cos(4t) = x^4 - 4x^2 + 2. - _Anthony C Robin_, Jun 02 2004
%C A034807 Triangle of coefficients of expansion of Z_{nk} in terms of Z_k.
%C A034807 Row n has 1 + floor(n/2) terms. - _Emeric Deutsch_, Dec 25 2004
%C A034807 T(n,k) = number of k-matchings of the cycle C_n (n > 1). Example: T(6,2)=9 because the 2-matchings of the hexagon with edges a, b, c, d, e, f are ac, ad, ae, bd, be, bf, ce, cf and df. - _Emeric Deutsch_, Dec 25 2004
%C A034807 An example for the first comment: G=c(x), H=1/(x*c(x)) with c(x) the o.g.f. Catalan numbers A000108: (x*c(x))^n + (1/c(x))^n = L(n,-x)= Sum_{k=0..floor(n/2)} T(n,k)*(-x)^k.
%C A034807 This triangle also supplies the absolute values of the coefficients in the multiplication formulas for the Lucas numbers A000032.
%C A034807 From _L. Edson Jeffery_, Mar 19 2011: (Start)
%C A034807 This sequence is related to rhombus substitution tilings. A signed version of it (see A132460), formed as a triangle with interlaced zeros extending each row to n terms, begins as
%C A034807 {2}
%C A034807 {1, 0}
%C A034807 {1, 0, -2}
%C A034807 {1, 0, -3, 0}
%C A034807 {1, 0, -4, 0, 2}
%C A034807 {1, 0, -5, 0, 5, 0}
%C A034807 ....
%C A034807 For the n X n tridiagonal unit-primitive matrix G_(n,1) (n >= 2) (see the L. E. Jeffery link below), defined by
%C A034807 G_(n,1) =
%C A034807 (0 1 0 ... 0)
%C A034807 (1 0 1 0 ... 0)
%C A034807 (0 1 0 1 0 ... 0)
%C A034807 ...
%C A034807 (0 ... 0 1 0 1)
%C A034807 (0 ... 0 2 0),
%C A034807 Row n (i.e., {T(n,k)}, k=0..n) of the signed table gives the coefficients of its characteristic function: c_n(x) = Sum_{k=0..n} T(n,k)*x^(n-k) = 0. For example, let n=3. Then
%C A034807 G_(3,1) =
%C A034807 (0 1 0)
%C A034807 (1 0 1)
%C A034807 (0 2 0),
%C A034807 and row 3 of the table is {1,0,-3,0}. Hence c_3(x) = x^3 - 3*x = 0. G_(n,1) has n distinct eigenvalues (the solutions of c_n(x) = 0), given by w_j = 2*cos((2*j-1)*Pi/(2*n)), j=1..n. (End)
%C A034807 For n > 0, T(n,k) is the number of k-subsets of {1,2,...,n} which contain neither consecutive integers nor both 1 and n. Equivalently, T(n,k) is the number of k-subsets without neighbors of a set of n points on a circle. - _José H. Nieto S._, Jan 17 2012
%C A034807 With the first column omitted, this gives A157000. - _Philippe Deléham_, Mar 17 2013
%C A034807 The number of necklaces of k black and n - k white beads with no adjacent black beads (Kaplansky 1943). Coefficients of the Dickson polynomials D(n,x,-a). - _Peter Bala_, Mar 09 2014
%C A034807 From _Tom Copeland_, Nov 07 2015: (Start)
%C A034807 This triangular array is composed of interleaved rows of reversed, unsigned A127677 (cf. A156308, A217476, A263916) and reversed A111125 (cf. A127672).
%C A034807 See also A113279 for another connection to symmetric and Faber polynomials.
%C A034807 The difference of consecutive rows gives the previous row shifted.
%C A034807 For relations among the characteristic polynomials of Cartan matrices of the Coxeter root groups, Chebyshev polynomials, cyclotomic polynomials, and the polynomials of this entry, see Damianou (p. 12, 20, and 21) and Damianou and Evripidou (p. 7). (End)
%C A034807 Diagonals are related to multiplicities of eigenvalues of the Laplacian on hyperspheres through A029635. - _Tom Copeland_, Jan 10 2016
%C A034807 For n>=3, also the independence and matching polynomials of the n-cycle graph C_n. See also A284966. - _Eric W. Weisstein_, Apr 06 2017
%C A034807 Apparently, with the rows aerated and then the 2s on the diagonal removed, this matrix becomes the reverse, or mirror, of unsigned A117179. See also A114525 - _Tom Copeland_, May 30 2017
%C A034807 Briggs's (1633) table with an additional column of 2s on the right can be used to generate this table. See p. 69 of the Newton reference. - _Tom Copeland_, Jun 03 2017
%C A034807 From _Liam Solus_, Aug 23 2018: (Start)
%C A034807 For n>3 and k>0, T(n,k) equals the number of Markov equivalence classes with skeleton the cycle on n nodes having exactly k immoralities. See Theorem 2.1 of the article by A. Radhakrishnan et al. below.
%C A034807 For n>2 odd and r = floor(n/2)-1, the n-th row is the coefficient vector of the Ehrhart h*-polynomial of the r-stable (n,2)-hypersimplex. See Theorem 4.14 in the article by B. Braun and L. Solus below.
%C A034807 (End)
%C A034807 Conjecture: If a(n) = H(a,b,c,d,n) is a second-order linear recurrence with constant coefficients defined as a(0) = a, a(1)= b, a(n) = c*a(n-1) + d*a(n-2) then a(m*n) = H(a, H(a,b,c,d,m), Sum_{k=0..floor(m/2)} T(m,k)*c^(m-2*k)*d^k, (-1)^(m+1)*d^m, n) (Wolfdieter Lang). - _Gary Detlefs_, Feb 06 2023
%C A034807 For the proof of the preceding conjecture see the Detlefs and Lang link. There also proofs for several properties of this table are found. - _Wolfdieter Lang_, Apr 25 2023
%C A034807 From _Mohammed Yaseen_, Nov 09 2024: (Start)
%C A034807 Let m - 1/m = x, then
%C A034807 m^2 + 1/m^2 = x^2 + 2,
%C A034807 m^3 - 1/m^3 = x^3 + 3*x,
%C A034807 m^4 + 1/m^4 = x^4 + 4*x^2 + 2,
%C A034807 m^5 - 1/m^5 = x^5 + 5*x^3 + 5*x,
%C A034807 m^6 + 1/m^6 = x^6 + 6*x^4 + 9*x^2 + 2,
%C A034807 m^7 - 1/m^7 = x^7 + 7*x^5 + 14*x^3 + 7*x, etc. (End)
%D A034807 A. Brousseau, Fibonacci and Related Number Theoretic Tables. Fibonacci Association, San Jose, CA, 1972, p. 148.
%D A034807 C. D. Godsil, Algebraic Combinatorics, Chapman and Hall, New York, 1993.
%D A034807 Thomas Koshy, Fibonacci and Lucas Numbers with Applications. New York, etc.: John Wiley & Sons, 2001. (Chapter 13, "Pascal-like Triangles," is devoted to the present triangle.)
%D A034807 The Royal Society Newton Tercentenary Celebrations, Cambridge Univ. Press, 1947.
%H A034807 T. D. Noe, <a href="/A034807/b034807.txt">Rows n = 0..100 of triangle, flattened</a>
%H A034807 Moussa Benoumhani, <a href="http://www.cs.uwaterloo.ca/journals/JIS//VOL6/Benoumhani/benoumhani8.html">A Sequence of Binomial Coefficients Related to Lucas and Fibonacci Numbers</a>, J. Integer Seqs., Vol. 6, 2003.
%H A034807 Benjamin Braun and Liam Solus, <a href="https://arxiv.org/abs/1408.4713">r-stable hypersimplices</a>, arXiv:1408.4713 [math.CO], 2014-2016; Journal of Combinatorial Theory, Series A 157 (2018): 349-388.
%H A034807 Johann Cigler and Hans-Christian Herbig, <a href="https://arxiv.org/abs/2412.18958">Factorization of spread polynomials</a>, arXiv:2412.18958 [math.NT], 2024. See p. 2.
%H A034807 Tom 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 A034807 Pantelis A. 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 A034807 Pantelis A. Damianou and Charalampos A. Evripidou, <a href="http://arxiv.org/abs/1409.3956">Characteristic and Coxeter polynomials for affine Lie algebras</a>, arXiv preprint arXiv:1409.3956 [math.RT], 2014.
%H A034807 Gary Detlefs and Wolfdieter Lang, <a href="https://arxiv.org/abs/2304.12937">Improved Formula for the Multi-Section of the Linear Three-Term Rcurrence Sequence<</a>, arXiv:2304.12937[nath.CO], 2023.
%H A034807 Joseph Doolittle, Lukas Katthän, Benjamin Nill, and Francisco Santos, <a href="https://arxiv.org/abs/2103.14925">Empty simplices of large width</a>, arXiv:2103.14925 [math.CO], 2021.
%H A034807 Sergio Falcon, <a href="http://scik.org/index.php/jmcs/article/view/102">On the Lucas triangle and its relationship with the k-Lucas numbers</a>, Journal of Mathematical and Computational Science, 2 (2012), No. 3, 425-434.
%H A034807 E. J. Farrell, <a href="http://dx.doi.org/10.1016/0095-8956(79)90070-4">An introduction to matching polynomials</a>, J. Combin. Theory B 27 (1) (1979), 75-86, Table 2.
%H A034807 Heidi Goodson, <a href="https://arxiv.org/abs/1901.08653">An Identity for Vertically Aligned Entries in Pascal's Triangle</a>, arXiv:1901.08653 [math.CO], 2019.
%H A034807 Gábor 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 A034807 L. E. Jeffery, <a href="https://oeis.org/wiki/User:L._Edson_Jeffery/Unit-Primitive_Matrices">Unit-primitive matrices</a>
%H A034807 I. Kaplansky, <a href="http://projecteuclid.org/euclid.bams/1183505432">Solution of the "probleme des menages"</a>, Bull. Amer. Math. Soc. 49, (1943). 784-785.
%H A034807 J. Kappraff and G. Adamson, <a href="http://vismath7.tripod.com/proceedings/kappraff.htm">Polygons and Chaos</a>, 5th Interdispl Symm. Congress and Exh. Jul 8-14, Sydney, 2001 - [with commercial pop-ups].
%H A034807 Emrah Kilic and Elif Tan Kilic, <a href="http://ekilic.etu.edu.tr/list/62Subseq.pdf">Some subsequences of the generalized Fibonacci and Lucas sequences</a>, Preprint, 2011.
%H A034807 Feihu Liu, Ying Wang, Yingrui Zhang, and Zihao Zhang,<a href="https://arxiv.org/abs/2503.17187">Hankel Determinants for Convolution of Power Series: An Extension of Cigler's Results</a>, arXiv:2503.17187 [math.CO], 2025. See p.5.
%H A034807 Eric Marberg, <a href="https://arxiv.org/abs/1709.07446">On some actions of the 0-Hecke monoids of affine symmetric groups</a>, arXiv:1709.07996 [math.CO], 2017.
%H A034807 T. J. Osler, <a href="http://www.jstor.org/stable/2691150">Cardan polynomials and the reduction of radicals</a>, Math. Mag., 74 (No. 1, 2001), 26-32.
%H A034807 Adityanarayanan Radhakrishnan, Liam Solus, and Caroline Uhler, <a href="https://arxiv.org/abs/1706.06091">Counting Markov equivalence classes for DAG models on trees</a>, Discrete Applied Mathematics 244 (2018): 170-185.
%H A034807 Lorenzo Venturello, <a href="https://arxiv.org/abs/2106.05051">Koszul Gorenstein algebras from Cohen-Macaulay simplicial complexes</a>, arXiv:2106.05051 [math.AC], 2021.
%H A034807 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/CycleGraph.html">Cycle Graph</a>
%H A034807 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/LucasPolynomial.html">Lucas Polynomial</a>
%H A034807 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/Matching-GeneratingPolynomial.html">Matching-Generating Polynomial</a>
%H A034807 Wikipedia, <a href="http://en.wikipedia.org/wiki/Dickson_polynomial">Dickson polynomial</a>.
%H A034807 Wikipedia, <a href="https://en.wikipedia.org/wiki/Metallic_mean">Metallic mean</a>.
%H A034807 Justin Eze Woko, <a href="https://doi.org/10.2307/3618772">A Pascal-like Triangle for alpha^n + beta^n</a>, The Mathematical Gazette, Vol. 81, No. 490 (Mar., 1997), pp. 75-79.
%F A034807 Row sums = A000032. T(2n, n-1) = A000290(n), T(2n+1, n-1) = A000330(n), T(2n, n-2) = A002415(n). T(n, k) = A029635(n-k, k), if n>0. - _Michael Somos_, Apr 02 1999
%F A034807 Lucas polynomial coefficients: 1, -n, n*(n-3)/2!, -n*(n-4)*(n-5)/3!, n*(n-5)*(n-6)*(n-7)/4!, - n*(n-6)*(n-7)*(n-8)*(n-9)/5!, ... - _Herb Conn_ and _Gary W. Adamson_, May 28 2003
%F A034807 G.f.: (2-x)/(1-x-x^2*y). - _Vladeta Jovovic_, May 31 2003
%F A034807 T(n, k) = T(n-1, k) + T(n-2, k-1), n>1. T(n, 0) = 1, n>0. T(n, k) = binomial(n-k, k) + binomial(n-k-1, k-1) = n*binomial(n-k-1, k-1)/k, 0 <= 2*k <= n except T(0, 0) = 2. - _Michael Somos_, Apr 02 1999
%F A034807 T(n,k) = (n*(n-1-k)!)/(k!*(n-2*k)!), n>0, k>=0. - Alexander Elkins (alexander_elkins(AT)hotmail.com), Jun 09 2007
%F A034807 O.g.f.: 2-(2xt+1)xt/(-t+xt+(xt)^2). (Cf. A113279.) - _Tom Copeland_, Nov 07 2015
%F A034807 T(n,k) = A011973(n-1,k) + A011973(n-3,k-1) = A011973(n,k) - A011973(n-4,k-2) except for T(0,0)=T(2,1)=2. - _Xiangyu Chen_, Dec 24 2020
%F A034807 L_n(x) = ((x+sqrt(x^2+4))/2)^n + (-((x+sqrt(x^2+4))/2))^(-n). See metallic means. - _William Krier_, Sep 01 2023
%e A034807 I have seen two versions of these polynomials: One version begins L_0 = 2, L_1 = 1, L_2 = 1 + 2*x, L_3 = 1 + 3*x, L_4 = 1 + 4*x + 2*x^2, L_5 = 1 + 5*x + 5*x^2, L_6 = 1 + 6*x + 9*x^2 + 2*x^3, L_7 = 1 + 7*x + 14*x^2 + 7*x^3, L_8 = 1 + 8*x + 20*x^2 + 16*x^3 + 2*x^4, L_9 = 1 + 9*x + 27*x^2 + 30*x^3 + 9*x^4, ...
%e A034807 The other version (probably the more official one) begins L_0(x) = 2, L_1(x) = x, L_2(x) = 2 + x^2, L_3(x) = 3*x + x^3, L_4(x) = 2 + 4*x^2 + x^4, L_5(x) = 5*x + 5*x^3 + x^5, L_6(x) = 2 + 9*x^2 + 6*x^4 + x^6, L_7(x) = 7*x + 14*x^3 + 7*x^5 + x^7, L_8(x) = 2 + 16*x^2 + 20*x^4 + 8*x^6 + x^8, L_9(x) = 9*x + 30*x^3 + 27*x^5 + 9*x^7 + x^9.
%e A034807 From _John Blythe Dobson_, Oct 11 2007: (Start)
%e A034807 Triangle begins:
%e A034807 2;
%e A034807 1;
%e A034807 1, 2;
%e A034807 1, 3;
%e A034807 1, 4, 2;
%e A034807 1, 5, 5;
%e A034807 1, 6, 9, 2;
%e A034807 1, 7, 14, 7;
%e A034807 1, 8, 20, 16, 2;
%e A034807 1, 9, 27, 30, 9;
%e A034807 1, 10, 35, 50, 25, 2;
%e A034807 1, 11, 44, 77, 55, 11;
%e A034807 1, 12, 54, 112, 105, 36, 2;
%e A034807 1, 13, 65, 156, 182, 91, 13;
%e A034807 1, 14, 77, 210, 294, 196, 49, 2;
%e A034807 1, 15, 90, 275, 450, 378, 140, 15;
%e A034807 (End)
%e A034807 From _Peter Bala_, Mar 20 2025: (Start)
%e A034807 Let S = x + y and M = -x*y. Then the triangle gives the coefficients when expressing the symmetric polynomial x^n + y^n as a polynomial in S and M. For example,
%e A034807 x^2 + y^2 = S^2 + 2*M; x^3 + y^3 = S^3 + 3*S*M; x^4 + y^4 = S^4 + 4*(S^2)*M + 2*M^2;
%e A034807 x^5 + y^5 = S^5 + 5*(S^3)*M + 5*S*M^2; x^6 + y^6 = S^6 + 6*(S^4)*M + 9*(S^2)*M^2 + 2*M^3. See Woko. In general x^n + y^n = 2*(-i)^n *(sqrt(M))^n * T(n, i*S/(2*sqrt(M))), where T(n, x) denotes the n-th Chebyshev polynomial of the first kind. (End)
%p A034807 T:= proc(n,k) if n=0 and k=0 then 2 elif k>floor(n/2) then 0 else n*binomial(n-k,k)/(n-k) fi end: for n from 0 to 15 do seq(T(n,k), k=0..floor(n/2)) od; # yields sequence in triangular form # _Emeric Deutsch_, Dec 25 2004
%t A034807 t[0, 0] = 2; t[n_, k_] := Binomial[n-k, k] + Binomial[n-k-1, k-1]; Table[t[n, k], {n, 0, 16}, {k, 0, Floor[n/2]}] // Flatten (* _Jean-François Alcover_, Dec 30 2013 *)
%t A034807 CoefficientList[Table[x^(n/2) LucasL[n, 1/Sqrt[x]], {n, 0, 15}], x] // Flatten (* _Eric W. Weisstein_, Apr 06 2017 *)
%t A034807 Table[Select[Reverse[CoefficientList[LucasL[n, x], x]], 0 < # &], {n, 0, 16}] // Flatten (* _Robert G. Wilson v_, May 03 2017 *)
%t A034807 CoefficientList[FunctionExpand @ Table[2 (-x)^(n/2) Cos[n ArcSec[2 Sqrt[-x]]], {n, 0, 15}], x] // Flatten (* _Eric W. Weisstein_, Apr 03 2018 *)
%t A034807 CoefficientList[Table[2 (-x)^(n/2) ChebyshevT[n, 1/(2 Sqrt[-x])], {n, 0, 15}], x] // Flatten (* _Eric W. Weisstein_, Apr 03 2018 *)
%o A034807 (PARI) {T(n, k) = if( k<0 || 2*k>n, 0, binomial(n-k, k) + binomial(n-k-1, k-1) + (n==0))}; /* _Michael Somos_, Jul 15 2003 */
%Y A034807 Cf. A029635, A061896, A111125, A113279, A114525, A127672, A127677, A156308, A217476, A263916.
%Y A034807 Cf. A117179.
%K A034807 nonn,tabf,easy
%O A034807 0,1
%A A034807 _N. J. A. Sloane_
%E A034807 Improved description, more terms, etc., from _Michael Somos_