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 A029635 #213 Feb 06 2025 23:44:47
%S A029635 2,1,2,1,3,2,1,4,5,2,1,5,9,7,2,1,6,14,16,9,2,1,7,20,30,25,11,2,1,8,27,
%T A029635 50,55,36,13,2,1,9,35,77,105,91,49,15,2,1,10,44,112,182,196,140,64,17,
%U A029635 2,1,11,54,156,294,378,336,204,81,19,2,1,12,65,210,450,672,714,540,285,100
%N A029635 The (1,2)-Pascal triangle (or Lucas triangle) read by rows.
%C A029635 This is also called Vieta's array. - _N. J. A. Sloane_, Nov 22 2017
%C A029635 Dropping the first term and changing the boundary conditions to T(n,1)=n, T(n,n-1)=2 (n>=2), T(n,n)=1 yields the number of nonterminal symbols (which generate strings of length k) in a certain context-free grammar in Chomsky normal form that generates all permutations of n symbols. Summation over k (1<=k<=n) results in A003945. For the number of productions of this grammar: see A090327. Example: 1; 2, 1; 3, 2, 1; 4, 5, 2, 1; 5, 9, 7, 2, 1; 6, 14, 16, 9, 2, 1; In addition to the example of A090327 we have T(3,3)=#{S}=1, T(3,2)=#{D,E}=2 and T(3,1)=#{A,B,C}=3. - _Peter R. J. Asveld_, Jan 29 2004
%C A029635 Much as the original Pascal triangle gives the Fibonacci numbers as sums of its diagonals, this triangle gives the Lucas numbers (A000032) as sums of its diagonals; see Posamentier & Lehmann (2007). - _Alonso del Arte_, Apr 09 2012
%C A029635 For a closed-form formula for generalized Pascal's triangle see A228576. - _Boris Putievskiy_, Sep 04 2013
%C A029635 It appears that for the infinite set of (1,N) Pascal's triangles, the binomial transform of the n-th row (n>0), followed by zeros, is equal to the n-th partial sum of (1, N, N, N, ...). Example: for the (1,2) Pascal's triangle, the binomial transform of the second row followed by zeros, i.e., of (1, 3, 2, 0, 0, 0, ...), is equal to the second partial sum of (1, 2, 2, 2, ...) = (1, 4, 9, 16, ...). - _Gary W. Adamson_, Aug 11 2015
%C A029635 Given any (1,N) Pascal triangle, let the binomial transform of the n-th row (n>1) followed by zeros be Q(x). It appears that the binomial transform of the (n-1)-th row prefaced by a zero is Q(n-1). Example: In the (1,2) Pascal triangle the binomial transform of row 3: (1, 4, 5, 2, 0, 0, 0, ...) is A000330 starting with 1: (1, 5, 14, 30, 55, 91, ...). The binomial transform of row 2 prefaced by a zero and followed by zeros, i.e., of (0, 1, 3, 2, 0, 0, 0, ...) is (0, 1, 5, 14, 30, 55, ...). - _Gary W. Adamson_, Sep 28 2015
%C A029635 It appears that in the array accompanying each (1,N) Pascal triangle (diagonals of the triangle), the binomial transform of (..., 1, N, 0, 0, 0, ...) preceded by (n-1) zeros generates the n-th row of the array (n>0). Then delete the zeros in the result. Example: in the (1,2) Pascal triangle, row 3 (1, 5, 14, 30, ...) is the binomial transform of (0, 0, 1, 2, 0, 0, 0, ...) with the resulting zeros deleted. - _Gary W. Adamson_, Oct 11 2015
%C A029635 Read as a square array (similar to the Example section Sq(m,j), but with Sq(0,0)=0 and Sq(m,j)=P(m+1,j) otherwise), P(n,k) are the multiplicities of the eigenvalues, lambda_n = n(n+k-1), of the Laplacians on the unit k-hypersphere, given by Teo (and Choi) as P(n,k) = (2n-k+1)(n+k-2)!/(n!(k-1)!). P(n,k) is also the numerator of a Dirichlet series for the Minakashisundarum-Pleijel zeta function for the sphere. Also P(n,k) is the dimension of the space of homogeneous, harmonic polynomials of degree k in n variables (Shubin, p. 169). For relations to Chebyshev polynomials and simple Lie algebras, see A034807. - _Tom Copeland_, Jan 10 2016
%C A029635 For a relation to a formulation for a universal Lie Weyl algebra for su(1,1), see page 16 of Durov et al. - _Tom Copeland_, Jan 15 2016
%D A029635 Boris A. Bondarenko, Generalized Pascal Triangles and Pyramids (in Russian), FAN, Tashkent, 1990, ISBN 5-648-00738-8.
%D A029635 Alfred S. Posamentier & Ingmar Lehmann, The (Fabulous) Fibonacci Numbers. New York: Prometheus Books (2007): 97 - 105.
%D A029635 M. Shubin and S. Andersson, Pseudodifferential Operators and Spectral Theory, Springer Series in Soviet Mathematics, 1987.
%H A029635 Vincenzo Librandi, <a href="/A029635/b029635.txt">Rows n = 0..102, flattened</a>
%H A029635 P. R. J. Asveld, <a href="http://dx.doi.org/10.1016/j.tcs.2005.11.010">Generating all permutations by context-free grammars in Chomsky normal form</a>, Theoretical Computer Science 354 (2006) 118-130.
%H A029635 Mohammad K. Azarian, <a href="http://www.m-hikari.com/ijcms/ijcms-2012/45-48-2012/azarianIJCMS45-48-2012.pdf">Identities Involving Lucas or Fibonacci and Lucas Numbers as Binomial Sums</a>, International Journal of Contemporary Mathematical Sciences, Vol. 7, No. 45, 2012, pp. 2221-2227.
%H A029635 Paul Barry, <a href="https://arxiv.org/abs/2004.04577">On a Central Transform of Integer Sequences</a>, arXiv:2004.04577 [math.CO], 2020.
%H A029635 Arthur T. Benjamin, <a href="http://www.math.hmc.edu/~benjamin/papers/LucasTriangle.pdf">The Lucas triangle recounted</a>
%H A029635 Boris A. Bondarenko, <a href="https://web.archive.org/web/2024*/https://www.fq.math.ca/pascal.html">Generalized Pascal Triangles and Pyramids</a>, English translation published by Fibonacci Association, Santa Clara Univ., Santa Clara, CA, 1993; see p. 25.
%H A029635 Junesang Choi, <a href="http://dx.doi.org/10.1186/1687-1847-2013-236">Determinants of the Laplacians on the n-dimensional unit sphere S^n </a>, Advances in Difference Equations, 236, 2013.
%H A029635 N. Durov, S. Meljanac, A. Samsarov, Z. Skoda, <a href="https://arxiv.org/abs/math/0604096">A universal formula for representing Lie algebra generators as formal power series with coefficients in the Weyl algebra</a>, arXiv preprint arXiv:math/0604096 [math.RT], 2006.
%H A029635 S. 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. - _N. J. A. Sloane_, Sep 20 2012
%H A029635 Mark Feinberg, <a href="https://web.archive.org/web/2024*/https://www.fq.math.ca/Scanned/5-5/feinberg.pdf">A Lucas triangle</a>, Fibonacci Quart. 5 (1967), 486-490.
%H A029635 Hans-Christian Herbig, Daniel Herden, Christopher Seaton, <a href="http://arxiv.org/abs/1605.01572">An algebra generated by x-2</a>, arXiv:1605.01572 [math.CO], 2016.
%H A029635 Milan Janjić, <a href="https://arxiv.org/abs/1905.04465">On Restricted Ternary Words and Insets</a>, arXiv:1905.04465 [math.CO], 2019.
%H A029635 M. A. A. Obaid, S. K. Nauman, W. M. Fakieh, C. M. Ringel, <a href="http://www.math.uni-bielefeld.de/~ringel/opus/jeddah.pdf">The numbers of support-tilting modules for a Dynkin algebra</a>, 2014 and <a href="https://cs.uwaterloo.ca/journals/JIS/VOL18/Ringel/ringel22.html">J. Int. Seq. 18 (2015) 15.10.6</a>.
%H A029635 Richard L. Ollerton and Anthony G. Shannon, <a href="https://web.archive.org/web/2024*/https://www.fq.math.ca/Scanned/36-2/ollerton.pdf">Some properties of generalized Pascal squares and triangles</a>, Fib. Q., 36 (1998), 98-109. [Here the initial term is 1 not 2]
%H A029635 C. M. Ringel, <a href="http://arxiv.org/abs/1502.06553">The Catalan combinatorics of the hereditary artin algebras</a>, arXiv preprint arXiv:1502.06553 [math.RT], 2015.
%H A029635 Neville Robbins, <a href="https://web.archive.org/web/2024*/https://www.fq.math.ca/Papers1/43-2/paper43-2-9.pdf">The Lucas triangle revisited</a>, Fibonacci Quarterly 43, May 2005, pp. 142-148.
%H A029635 Lee-Peng Teo, <a href="http://arxiv.org/abs/1412.0758">Zeta function of spheres and real projective spaces</a>, arXiv:1412.0758v1 [math.NT], 2014.
%H A029635 Xu Wang, Xuxu Zhao, Haiyuan Yao, <a href="https://arxiv.org/abs/1810.07329">Structure and enumeration results of matchable Lucas cubes</a>, arXiv:1810.07329 [math.CO], 2018. See Table 1 p. 3.
%H A029635 Y. Yang and J. Leida, <a href="https://web.archive.org/web/2024*/https://www.fq.math.ca/Papers1/42-3/quartyang03_2004.pdf">Pascal decompositions of geometric arrays in matrices</a>, The Fibonacci Quarterly 42.3 (2004).
%F A029635 From _Henry Bottomley_, Apr 26 2002; (Start)
%F A029635 T(n, k) = T(n-1, k-1) + T(n-1, k).
%F A029635 T(n, k) = C(n, k) + C(n-1, k-1).
%F A029635 T(n, k) = C(n, k)*(n + k)/n.
%F A029635 T(n, k) = A007318(n, k) + A007318(n-1, k-1).
%F A029635 T(n, k) = A061896(n + k, k) but with T(0, 0) = 1 and T(1, 1) = 2.
%F A029635 Row sum is floor(3^2(n-1)) i.e., A003945. (End)
%F A029635 G.f.: 1 + (1 + x*y) / (1 - x - x*y). - _Michael Somos_, Jul 15 2003
%F A029635 G.f. for n-th row: (x+2*y)*(x+y)^(n-1).
%F A029635 O.g.f. for row n: (1+x)/(1-x)^(n+1). The entries in row n are the nonzero entries in column n of A053120 divided by 2^(n-1). - _Peter Bala_, Aug 14 2008
%F A029635 T(2n, n) - T(2n, n+1)= Catalan(n)= A000108(n). - _Philippe Deléham_, Mar 19 2009
%F A029635 With T(0, 0) = 1 : Triangle T(n, k), read by rows, given by [1,0,0,0,0,0,...] DELTA [2,-1,0,0,0,0,...] where DELTA is the operator defined in A084938. - _Philippe Deléham_, Oct 10 2011
%F A029635 With T(0, 0) = 1, as in the Example section below, this is known as Vieta's array. The LU factorization of the square array is given by Yang and Leida, equation 20. - _Peter Bala_, Feb 11 2012
%F A029635 For n > 0: T(n, k) = A097207(n-1, k), 0 <= k < n. - _Reinhard Zumkeller_, Mar 12 2012
%F A029635 For n > 0: T(n, k) = A029600(n, k) - A007318(n, k), 0 <= k <= n. - _Reinhard Zumkeller_, Apr 16 2012
%F A029635 Riordan array ((2-x)/(1-x), x/(1-x)). - _Philippe Deléham_, Mar 15 2013
%F A029635 exp(x) * e.g.f. for row n = e.g.f. for diagonal n. For example, for n = 3 we have exp(x)*(1 + 4*x + 5*x^2/2! + 2*x^3/3!) = 1 + 5*x + 14*x^2/2! + 30*x^3/3! + 55*x^4/4! + .... The same property holds more generally for Riordan arrays of the form ( f(x), x/(1 - x) ). - _Peter Bala_, Dec 22 2014
%F A029635 For n>=1: T(n, 0) + T(n, 1) + T(n, 2) = A000217(n+1). T(n, n-2) = (n-1)^2. - _Bob Selcoe_, Mar 29 2016:
%e A029635 Triangle begins:
%e A029635 [0] [2]
%e A029635 [1] [1, 2]
%e A029635 [2] [1, 3, 2]
%e A029635 [3] [1, 4, 5, 2]
%e A029635 [4] [1, 5, 9, 7, 2]
%e A029635 [5] [1, 6, 14, 16, 9, 2]
%e A029635 [6] [1, 7, 20, 30, 25, 11, 2]
%e A029635 [7] [1, 8, 27, 50, 55, 36, 13, 2]
%e A029635 [8] [1, 9, 35, 77, 105, 91, 49, 15, 2]
%e A029635 .
%e A029635 Read as a square, the array begins:
%e A029635 n\k| 0 1 2 3 4 5
%e A029635 --------------------------------------
%e A029635 0 | 2 2 2 2 2 2 A040000
%e A029635 1 | 1 3 5 7 9 11 A005408
%e A029635 2 | 1 4 9 16 25 36 A000290
%e A029635 3 | 1 5 14 30 55 91 A000330
%e A029635 4 | 1 6 20 50 105 196 A002415
%e A029635 5 | 1 7 27 77 182 378 A005585
%e A029635 6 | 1 8 35 112 294 672 A040977
%p A029635 T := proc(n, k) option remember;
%p A029635 if n = k then 2 elif k = 0 then 1 else T(n-1, k-1) + T(n-1, k) fi end:
%p A029635 for n from 0 to 8 do seq(T(n, k), k = 0..n) od; # _Peter Luschny_, Dec 22 2024
%t A029635 t[0, 0] = 2; t[n_, k_] := If[k < 0 || k > n, 0, Binomial[n, k] + Binomial[n-1, k-1]]; Flatten[Table[t[n, k], {n, 0, 11}, {k, 0, n}]] (* _Jean-François Alcover_, May 03 2011 *)
%t A029635 (* The next program cogenerates A029635 and A029638. *)
%t A029635 u[1, x_] := 1; v[1, x_] := 1; z = 16;
%t A029635 u[n_, x_] := u[n - 1, x] + v[n - 1, x]
%t A029635 v[n_, x_] := x*u[n - 1, x] + x*v[n - 1, x] + 1
%t A029635 Table[Factor[u[n, x]], {n, 1, z}]
%t A029635 Table[Factor[v[n, x]], {n, 1, z}]
%t A029635 cu = Table[CoefficientList[u[n, x], x], {n, 1, z}];
%t A029635 TableForm[cu]
%t A029635 Flatten[%] (* A029638 *)
%t A029635 Table[Expand[v[n, x]], {n, 1, z}]
%t A029635 cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
%t A029635 TableForm[cv]
%t A029635 Flatten[%] (* A029635 *)
%t A029635 (* _Clark Kimberling_, Feb 20 2012 *)
%t A029635 Table[Binomial[n,k]+Binomial[n-1,k-1],{n,0,20},{k,0,n}]//Flatten (* _Harvey P. Dale_, Feb 08 2024 *)
%o A029635 (PARI) {T(n, k) = if( k<0 || k>n, 0, (n==0) + binomial(n, k) + binomial(n-1, k-1))}; /* _Michael Somos_, Jul 15 2003 */
%o A029635 (Haskell)
%o A029635 a029635 n k = a029635_tabl !! n !! k
%o A029635 a029635_row n = a029635_tabl !! n
%o A029635 a029635_tabl = [2] : iterate
%o A029635 (\row -> zipWith (+) ([0] ++ row) (row ++ [0])) [1,2]
%o A029635 -- _Reinhard Zumkeller_, Mar 12 2012, Feb 23 2012
%o A029635 (Sage) # uses[riordan_array from A256893]
%o A029635 riordan_array((2-x)/(1-x), x/(1-x), 8) # _Peter Luschny_, Nov 09 2019
%Y A029635 Cf. A003945 (row sums), A007318, A034807, A061896, A029653 (row-reversed), A157000.
%Y A029635 Sums along ascending antidiagonals give Lucas numbers, n>0.
%Y A029635 Cf. A090327, A003945, A029638, A228196, A228576, A000330, A000217, A293600.
%K A029635 nonn,tabl,nice,easy
%O A029635 0,1
%A A029635 _Mohammad K. Azarian_
%E A029635 More terms from _David W. Wilson_
%E A029635 a(0) changed to 2 (was 1) by _Daniel Forgues_, Jul 06 2010