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%I #269 Mar 28 2024 21:54:21
%S 1,2,1,5,4,1,14,14,6,1,42,48,27,8,1,132,165,110,44,10,1,429,572,429,
%T 208,65,12,1,1430,2002,1638,910,350,90,14,1,4862,7072,6188,3808,1700,
%U 544,119,16,1,16796,25194,23256,15504,7752,2907,798,152,18,1
%N Triangle formed from odd-numbered columns of triangle of expansions of powers of x in terms of Chebyshev polynomials U_n(x). Sometimes called Catalan's triangle.
%C T(n,k) is the number of leaves at level k+1 in all ordered trees with n+1 edges. - _Emeric Deutsch_, Jan 15 2005
%C Riordan array ((1-2x-sqrt(1-4x))/(2x^2),(1-2x-sqrt(1-4x))/(2x)). Inverse array is A053122. - _Paul Barry_, Mar 17 2005
%C T(n,k) is the number of walks of n steps, each in direction N, S, W, or E, starting at the origin, remaining in the upper half-plane and ending at height k (see the _R. K. Guy_ reference, p. 5). Example: T(3,2)=6 because we have ENN, WNN, NEN, NWN, NNE and NNW. - _Emeric Deutsch_, Apr 15 2005
%C Triangle T(n,k), 0<=k<=n, read by rows given by T(0,0)=1, T(n,k)=0 if k<0 or if k>n, T(n,0) = 2*T(n-1,0) + T(n-1,1), T(n,k) = T(n-1,k-1) + 2*T(n-1,k) + T(n-1,k+1) for k>=1. - _Philippe Deléham_, Mar 30 2007
%C Number of (2n+1)-step walks from (0,0) to (2n+1,2k+1) and consisting of steps u=(1,1) and d=(1,-1) in which the path stays in the nonnegative quadrant. Examples: T(2,0)=5 because we have uuudd, uudud, uuddu, uduud, ududu; T(2,1)=4 because we have uuuud, uuudu, uuduu, uduuu; T(2,2)=1 because we have uuuuu. - _Philippe Deléham_, Apr 16 2007, Apr 18 2007
%C Triangle read by rows: T(n,k)=number of lattice paths from (0,0) to (n,k) that do not go below the line y=0 and consist of steps U=(1,1), D=(1,-1) and two types of steps H=(1,0); example: T(3,1)=14 because we have UDU, UUD, 4 HHU paths, 4 HUH paths and 4 UHH paths. - _Philippe Deléham_, Sep 25 2007
%C This triangle belongs to the family of triangles defined by T(0,0)=1, T(n,k)=0 if k<0 or if k>n, T(n,0) = x*T(n-1,0) + T(n-1,1), T(n,k) = T(n-1,k-1) + y*T(n-1,k) + T(n-1,k+1) for k>=1. Other triangles arise by choosing different values for (x,y): (0,0) -> A053121; (0,1) -> A089942; (0,2) -> A126093; (0,3) -> A126970; (1,0) -> A061554; (1,1) -> A064189; (1,2) -> A039599; (1,3) -> A110877; (1,4) -> A124576; (2,0) -> A126075; (2,1) -> A038622; (2,2) -> A039598; (2,3) -> A124733; (2,4) -> A124575; (3,0) -> A126953; (3,1) -> A126954; (3,2) -> A111418; (3,3) -> A091965; (3,4) -> A124574; (4,3) -> A126791; (4,4) -> A052179; (4,5) -> A126331; (5,5) -> A125906. - _Philippe Deléham_, Sep 25 2007
%C With offset [1,1] this is the (ordinary) convolution triangle a(n,m) with o.g.f. of column m given by (c(x)-1)^m, where c(x) is the o.g.f. for Catalan numbers A000108. See the Riordan comment by _Paul Barry_.
%C T(n, k) is also the number of order-preserving full transformations (of an n-chain) with exactly k fixed points. - _Abdullahi Umar_, Oct 02 2008
%C T(n,k)/2^(2n+1) = coefficients of the maximally flat lowpass digital differentiator of the order N=2n+3. - Pavel Holoborodko (pavel(AT)holoborodko.com), Dec 19 2008
%C The signed triangle S(n,k) := (-1)^(n-k)*T(n,k) provides the transformation matrix between f(n,l) := L(2*l)*5^n*F(2*l)^(2*n+1) (F=Fibonacci numbers A000045, L=Lucas numbers A000032) and F(4*l*(k+1)), k = 0, ..., n, for each l>=0: f(n,l) = Sum_{k=0..n} S(n,k)*F(4*l*(k+1)), n>=0, l>=0. Proof: the o.g.f. of the l.h.s., G(l;x) := Sum_{n>=0} f(n,l)*x^n = F(4*l)/(1 - 5*F(2*l)^2*x) is shown to match the o.g.f. of the r.h.s.: after an interchange of the n- and k-summation, the Riordan property of S = (C(x)/x,C(x)) (compare with the above comments by _Paul Barry_), with C(x) := 1 - c(-x), with the o.g.f. c(x) of A000108 (Catalan numbers), is used, to obtain, after an index shift, first Sum_{k>=0} F(4*l*(k))*GS(k;x), with the o.g.f of column k of triangle S which is GS(k;x) := Sum_{n>=k} S(n,k)*x^n = C(x)^(k+1)/x. The result is GF(l;C(x))/x with the o.g.f. GF(l,x) := Sum_{k>=0} F(4*l*k)*x^k = x*F(4*l)/(1-L(4*l)*x+x^2) (see a comment on A049670, and A028412). If one uses then the identity L(4*n) - 5*F(2*n)^2 = 2 (in Koshy's book [reference under A065563] this is No. 15, p. 88, attributed to Lucas, 1876), the proof that one recovers the o.g.f. of the l.h.s. from above boils down to a trivial identity on the Catalan o.g.f., namely 1/c^2(-x) = 1 + 2*x - (x*c(-x))^2. - _Wolfdieter Lang_, Aug 27 2012
%C O.g.f. for row polynomials R(x) := Sum_{k=0..n} a(n,k)*x^k:
%C ((1+x) - C(z))/(x - (1+x)^2*z) with C the o.g.f. of A000108 (Catalan numbers). From Riordan ((C(x)-1)/x,C(x)-1), compare with a _Paul Barry_ comment above. This coincides with the o.g.f. given by _Emeric Deutsch_ in the formula section. - _Wolfdieter Lang_, Nov 13 2012
%C The A-sequence for this Riordan triangle is [1,2,1] and the Z-sequence is [2,1]. See a W. Lang link under A006232 with details and references. - _Wolfdieter Lang_, Nov 13 2012
%C From _Wolfdieter Lang_, Sep 20 2013: (Start)
%C T(n, k) = A053121(2*n+1, 2*k+1). T(n, k) appears in the formula for the (2*n+1)-th power of the algebraic number rho(N) := 2*cos(Pi/N) = R(N, 2) in terms of the even-indexed diagonal/side length ratios R(N, 2*(k+1)) = S(2*k+1, rho(N)) in the regular N-gon inscribed in the unit circle (length unit 1). S(n, x) are Chebyshev's S polynomials (see A049310): rho(N)^(2*n+1) = Sum_{k=0..n} T(n, k)*R(N, 2*(k+1)), n >= 0, identical in N >= 1. For a proof see the Sep 21 2013 comment under A053121. Note that this is the unreduced version if R(N, j) with j > delta(N), the degree of the algebraic number rho(N) (see A055034), appears. For the even powers of rho(n) see A039599. (End)
%C The tridiagonal Toeplitz production matrix P in the Example section corresponds to the unsigned Cartan matrix for the simple Lie algebra A_n as n tends to infinity (cf. Damianou ref. in A053122). - _Tom Copeland_, Dec 11 2015 (revised Dec 28 2015)
%C T(n,k) is the number of pairs of non-intersecting walks of n steps, each in direction N or E, starting at the origin, and such that the end points of the two paths are separated by a horizontal distance of k. See Shapiro 1976. - _Peter Bala_, Apr 12 2017
%C Also the convolution triangle of the Catalan numbers A000108. - _Peter Luschny_, Oct 07 2022
%D M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 796.
%D B. A. Bondarenko, Generalized Pascal Triangles and Pyramids (in Russian), FAN, Tashkent, 1990, ISBN 5-648-00738-8.
%H G. C. Greubel, <a href="/A039598/b039598.txt">Table of n, a(n) for the first 50 rows, flattened</a>
%H M. Abramowitz and I. A. Stegun, eds., <a href="http://www.convertit.com/Go/ConvertIt/Reference/AMS55.ASP">Handbook of Mathematical Functions</a>, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
%H José Agapito, Ângela Mestre, Maria M. Torres, and Pasquale Petrullo, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL18/Agapito/agapito2.html">On One-Parameter Catalan Arrays</a>, Journal of Integer Sequences, Vol. 18 (2015), Article 15.5.1.
%H M. Aigner, <a href="http://dx.doi.org/10.1016/j.disc.2007.06.012">Enumeration via ballot numbers</a>, Discrete Math., 308 (2008), 2544-2563.
%H Quang T. Bach and Jeffrey B. Remmel, <a href="http://arxiv.org/abs/1510.04319">Generating functions for descents over permutations which avoid sets of consecutive patterns</a>, arXiv:1510.04319 [math.CO], 2015 (see p. 25).
%H Peter Bala, <a href="/A100100/a100100.pdf">Notes on logarithmic differentiation, the binomial transform and series reversion</a>
%H Jean-Luc Baril, José L. Ramírez, and Lina M. Simbaqueba, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL24/Ramirez/ramirez10.html">Counting prefixes of skew Dyck paths</a>, J. Int. Seq., Vol. 24 (2021), Article 21.8.2.
%H Paul Barry, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL15/Barry5/barry223.html">On the Hurwitz Transform of Sequences</a>, Journal of Integer Sequences, Vol. 15 (2012), #12.8.7.
%H Paul Barry, <a href="https://www.emis.de/journals/JIS/VOL22/Barry3/barry422.html">Generalized Catalan Numbers Associated with a Family of Pascal-like Triangles</a>, J. Int. Seq., Vol. 22 (2019), Article 19.5.8.
%H Paul Barry, <a href="https://arxiv.org/abs/1912.01124">A Note on Riordan Arrays with Catalan Halves</a>, arXiv:1912.01124 [math.CO], 2019.
%H Paul Barry, <a href="https://arxiv.org/abs/1912.11845">Chebyshev moments and Riordan involutions</a>, arXiv:1912.11845 [math.CO], 2019.
%H Paul Barry, <a href="https://arxiv.org/abs/2011.10827">Notes on the Hankel transform of linear combinations of consecutive pairs of Catalan numbers</a>, arXiv:2011.10827 [math.CO], 2020.
%H B. A. Bondarenko, <a href="http://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. 29.
%H Eduardo H. M. Brietzke, <a href="http://www.fq.math.ca/Papers1/44-2/quarteduardobrietzke02_2006.pdf">Generalization of an identity of Andrews</a>, Fibonacci Quart. 44 (2006), no. 2, 166-171.
%H F. Cai, Q.-H. Hou, Y. Sun, and A. L. B. Yang, <a href="http://arxiv.org/abs/1808.05736">Combinatorial identities related to 2x2 submatrices of recursive matrices</a>, arXiv:1808.05736 [math.CO], 2018. See Table 1.1.
%H Naiomi T. Cameron and Asamoah Nkwanta, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL8/Cameron/cameron46.html">On Some (Pseudo) Involutions in the Riordan Group</a>, Journal of Integer Sequences, Vol. 8 (2005), Article 05.3.7.
%H Marc Chamberland, <a href="https://doi.org/10.1016/j.laa.2011.08.030">Factored matrices can generate combinatorial identities</a>, Linear Algebra and its Applications, Volume 438, Issue 4, 15 Feb. 2013, pp. 1667-1677.
%H Xi Chen, H. Liang, and Y. Wang, <a href="http://arxiv.org/abs/1601.05645">Total positivity of recursive matrices</a>, arXiv:1601.05645 [math.CO], 2016.
%H Xi Chen, H. Liang, and Y. Wang, <a href="http://dx.doi.org/10.1016/j.laa.2015.01.009">Total positivity of recursive matrices</a>, Linear Algebra and its Applications, Volume 471, Apr 15 2015, Pages 383-393.
%H Johann Cigler, <a href="https://arxiv.org/abs/1611.05252">Some elementary observations on Narayana polynomials and related topics</a>, arXiv:1611.05252 [math.CO], 2016. See p. 7.
%H S. J. Cyvin, J. Brunvoll, E. Brendsdal, B. N. Cyvin and E. K. Lloyd, <a href="/A002057/a002057.pdf">Enumeration of polyene hydrocarbons: a complete mathematical solution</a>, J. Chem. Inf. Comput. Sci., 35 (1995) 743-751. [Annotated scanned copy]
%H Paul Drube, <a href="https://arxiv.org/abs/2007.01892">Generalized Path Pairs and Fuss-Catalan Triangles</a>, arXiv:2007.01892 [math.CO], 2020. See Figure 4 p. 8.
%H Shishuo Fu and Yaling Wang, <a href="https://arxiv.org/abs/1908.03912">Bijective recurrences concerning two Schröder triangles</a>, arXiv:1908.03912 [math.CO], 2019.
%H R. K. Guy, <a href="http://www.cs.uwaterloo.ca/journals/JIS/VOL3/GUY/catwalks.html">Catwalks, sandsteps and Pascal pyramids</a>, J. Integer Sequences, Vol. 3 (2000), Article #00.1.6.
%H T.-X. He and L. W. Shapiro, <a href="http://dx.doi.org/10.1016/j.laa.2017.06.025">Fuss-Catalan matrices, their weighted sums, and stabilizer subgroups of the Riordan group</a>, Lin. Alg. Applic. 532 (2017) 25-41, example page 32.
%H Peter M. Higgins, <a href="http://sci-prew.inf.ua/v113/2/S0305004100075964.pdf">Combinatorial results for semigroups of order-preserving mappings</a>, Math. Proc. Camb. Phil. Soc. 113 (1993), 281-296.
%H A. Laradji and A. Umar, <a href="http://dx.doi.org/10.1007/s00233-005-0553-6">Combinatorial results for semigroups of order-preserving full transformations</a>, Semigroup Forum 72 (2006), 51-62.
%H Donatella Merlini and Renzo Sprugnoli, <a href="https://doi.org/10.1016/j.disc.2016.08.017">Arithmetic into geometric progressions through Riordan arrays</a>, Discrete Mathematics 340.2 (2017): 160-174. See (1.1).
%H Pedro J. Miana, Hideyuki Ohtsuka, and Natalia Romero, <a href="http://arxiv.org/abs/1602.04347">Sums of powers of Catalan triangle numbers</a>, arXiv:1602.04347 [math.NT], 2016 (see 2.4).
%H Asamoah Nkwanta and Earl R. Barnes, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL15/Nkwanta/nkwanta2.html">Two Catalan-type Riordan Arrays and their Connections to the Chebyshev Polynomials of the First Kind</a>, Journal of Integer Sequences, Article 12.3.3, 2012. - From _N. J. A. Sloane_, Sep 16 2012
%H A. Nkwanta and A. Tefera, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL16/Nkwanta/nkwanta4.html">Curious Relations and Identities Involving the Catalan Generating Function and Numbers</a>, Journal of Integer Sequences, 16 (2013), #13.9.5.
%H L. W. Shapiro, W.-J. Woan and S. Getu, <a href="http://dx.doi.org/10.1137/0604046">Runs, slides and moments</a>, SIAM J. Alg. Discrete Methods, 4 (1983), 459-466.
%H L. W. Shapiro, <a href="http://dx.doi.org/10.1016/0012-365X(76)90009-1">A Catalan triangle</a>, Discrete Math., 14, 83-90, 1976.
%H L. W. Shapiro, <a href="/A003517/a003517.pdf">A Catalan triangle</a>, Discrete Math. 14 (1976), no. 1, 83-90. [Annotated scanned copy]
%H Yidong Sun and Fei Ma, <a href="http://arxiv.org/abs/1305.2015">Minors of a Class of Riordan Arrays Related to Weighted Partial Motzkin Paths</a>, arXiv preprint arXiv:1305.2015 [math.CO], 2013.
%H Yidong Sun and Fei Ma, <a href="http://arxiv.org/abs/1305.2017">Four transformations on the Catalan triangle</a>, arXiv preprint arXiv:1305.2017 [math.CO], 2013.
%H Yidong Sun and Fei Ma, <a href="http://www.combinatorics.org/ojs/index.php/eljc/article/view/v21i1p33">Some new binomial sums related to the Catalan triangle</a>, Electronic Journal of Combinatorics 21(1) (2014), #P1.33.
%H Paweł J. Szabłowski, <a href="https://arxiv.org/abs/2106.10461">Yet another way of calculating moments of the Kesten's distribution and its consequences</a>, arXiv:2106.10461 [math.CO], 2021.
%H Charles Zhao-Chen Wang and Yi Wang, <a href="http://dx.doi.org/10.1016/j.disc.2014.11.017">Total positivity of Catalan triangle</a>, Discrete Math. 338 (2015), no. 4, 566--568. MR3300743.
%H W.-J. Woan, L. Shapiro and D. G. Rogers, <a href="http://www.jstor.org/stable/2974473">The Catalan numbers, the Lebesgue integral and 4^{n-2}</a>, Amer. Math. Monthly, 104 (1997), 926-931.
%H Sheng-Liang Yang, Yan-Ni Dong, and Tian-Xiao He, <a href="https://doi.org/10.1016/j.disc.2017.07.006">Some matrix identities on colored Motzkin paths</a>, Discrete Mathematics 340.12 (2017): 3081-3091.
%F Row n: C(2n, n-k) - C(2n, n-k-2).
%F a(n, k) = C(2n+1, n-k)*2*(k+1)/(n+k+2) = A050166(n, n-k) = a(n-1, k-1) + 2*a(n-1, k)+ a (n-1, k+1) [with a(0, 0) = 1 and a(n, k) = 0 if n<0 or n<k]. - _Henry Bottomley_, Sep 24 2001
%F From _Philippe Deléham_, Feb 14 2004: (Start)
%F T(n, 0) = A000108(n+1), T(n, k) = 0 if n<k; for k>0, T(n, k) = Sum_{j=1..n} T(n-j, k-1)*A000108(j).
%F G.f. for column k: Sum_{n>=0} T(n, k)*x^n = x^k*C(x)^(2*k+2) where C(x) = Sum_{n>=0} A000108(n)*x^n is g.f. for Catalan numbers, A000108.
%F Sum_{k>=0} T(m, k)*T(n, k) = A000108(m+n+1). (End)
%F T(n, k) = A009766(n+k+1, n-k) = A033184(n+k+2, 2k+2). - _Philippe Deléham_, Feb 14 2004
%F Sum_{j>=0} T(k, j)*A039599(n-k, j) = A028364(n, k). - _Philippe Deléham_, Mar 04 2004
%F Antidiagonal Sum_{k=0..n} T(n-k, k) = A000957(n+3). - _Gerald McGarvey_, Jun 05 2005
%F The triangle may also be generated from M^n * [1,0,0,0,...], where M = an infinite tridiagonal matrix with 1's in the super- and subdiagonals and [2,2,2,...] in the main diagonal. - _Gary W. Adamson_, Dec 17 2006
%F G.f.: G(t,x) = C^2/(1-txC^2), where C = (1-sqrt(1-4x))/(2x) is the Catalan function. From here G(-1,x)=C, i.e., the alternating row sums are the Catalan numbers (A000108). - _Emeric Deutsch_, Jan 20 2007
%F Sum_{k=0..n} T(n,k)*x^k = A000957(n+1), A000108(n), A000108(n+1), A001700(n), A049027(n+1), A076025(n+1), A076026(n+1) for x=-2,-1,0,1,2,3,4 respectively (see square array in A067345). - _Philippe Deléham_, Mar 21 2007, Nov 04 2011
%F Sum_{k=0..n} T(n,k)*(k+1) = 4^n. - _Philippe Deléham_, Mar 30 2007
%F Sum_{j>=0} T(n,j)*binomial(j,k) = A035324(n,k), A035324 with offset 0 (0 <= k <= n). - _Philippe Deléham_, Mar 30 2007
%F T(n,k) = A053121(2*n+1,2*k+1). - _Philippe Deléham_, Apr 16 2007, Apr 18 2007
%F T(n,k) = A039599(n,k) + A039599(n,k+1). - _Philippe Deléham_, Sep 11 2007
%F Sum_{k=0..n+1} T(n+1,k)*k^2 = A029760(n). - _Philippe Deléham_, Dec 16 2007
%F Sum_{k=0..n} T(n,k)*A059841(k) = A000984(n). - _Philippe Deléham_, Nov 12 2008
%F G.f.: 1/(1-xy-2x-x^2/(1-2x-x^2/(1-2x-x^2/(1-2x-x^2/(1-2x-x^2/(1-.... (continued fraction).
%F Sum_{k=0..n} T(n,k)*x^(n-k) = A000012(n), A001700(n), A194723(n+1), A194724(n+1), A194725(n+1), A194726(n+1), A194727(n+1), A194728(n+1), A194729(n+1), A194730(n+1) for x = 0,1,2,3,4,5,6,7,8,9 respectively. - _Philippe Deléham_, Nov 03 2011
%F From _Peter Bala_, Dec 21 2014: (Start)
%F This triangle factorizes in the Riordan group as ( C(x), x*C(x) ) * ( 1/(1 - x), x/(1 - x) ) = A033184 * A007318, where C(x) = (1 - sqrt(1 - 4*x))/(2*x) is the o.g.f. for the Catalan numbers A000108.
%F Let U denote the lower unit triangular array with 1's on or below the main diagonal and zeros elsewhere. For k = 0,1,2,... define U(k) to be the lower unit triangular block array
%F /I_k 0\
%F \ 0 U/ having the k X k identity matrix I_k as the upper left block; in particular, U(0) = U. Then this array equals the bi-infinite product (...*U(2)*U(1)*U(0))*(U(0)*U(1)*U(2)*...). (End)
%F From _Peter Bala_, Jul 21 2015: (Start)
%F O.g.f. G(x,t) = (1/x) * series reversion of ( x/f(x,t) ), where f(x,t) = ( 1 + (1 + t)*x )^2/( 1 + t*x ).
%F 1 + x*d/dx(G(x,t))/G(x,t) = 1 + (2 + t)*x + (6 + 4*t + t^2)*x^2 + ... is the o.g.f for A094527. (End)
%F Conjecture: Sum_{k=0..n} T(n,k)/(k+1)^2 = H(n+1)*A000108(n)*(2*n+1)/(n+1), where H(n+1) = Sum_{k=0..n} 1/(k+1). - _Werner Schulte_, Jul 23 2015
%F From _Werner Schulte_, Jul 25 2015: (Start)
%F Sum_{k=0..n} T(n,k)*(k+1)^2 = (2*n+1)*binomial(2*n,n). (A002457)
%F Sum_{k=0..n} T(n,k)*(k+1)^3 = 4^n*(3*n+2)/2.
%F Sum_{k=0..n} T(n,k)*(k+1)^4 = (2*n+1)^2*binomial(2*n,n).
%F Sum_{k=0..n} T(n,k)*(k+1)^5 = 4^n*(15*n^2+15*n+4)/4. (End)
%F The o.g.f. G(x,t) is such that G(x,t+1) is the o.g.f. for A035324, but with an offset of 0, and G(x,t-1) is the o.g.f. for A033184, again with an offset of 0. - _Peter Bala_, Sep 20 2015
%F Denote this lower triangular array by L; then L * transpose(L) is the Cholesky factorization of the Hankel matrix ( 1/(i+j)*binomial(2*i+2*j-2, i+j-1) )_i,j >= 1 = A172417 read as a square array. See Chamberland, p. 1669. - _Peter Bala_, Oct 15 2023
%e Triangle T(n,k) starts:
%e n\k 0 1 2 3 4 5 6 7 8 9 10
%e 0: 1
%e 1: 2 1
%e 2: 5 4 1
%e 3: 14 14 6 1
%e 4: 42 48 27 8 1
%e 5: 132 165 110 44 10 1
%e 6: 429 572 429 208 65 12 1
%e 7: 1430 2002 1638 910 350 90 14 1
%e 8: 4862 7072 6188 3808 1700 544 119 16 1
%e 9: 16796 25194 23256 15504 7752 2907 798 152 18 1
%e 10: 58786 90440 87210 62016 33915 14364 4655 1120 189 20 1
%e ... Reformatted and extended by _Wolfdieter Lang_, Nov 13 2012.
%e Production matrix begins:
%e 2, 1
%e 1, 2, 1
%e 0, 1, 2, 1
%e 0, 0, 1, 2, 1
%e 0, 0, 0, 1, 2, 1
%e 0, 0, 0, 0, 1, 2, 1
%e 0, 0, 0, 0, 0, 1, 2, 1
%e 0, 0, 0, 0, 0, 0, 1, 2, 1
%e - _Philippe Deléham_, Nov 07 2011
%e From _Wolfdieter Lang_, Nov 13 2012: (Start)
%e Recurrence: T(5,1) = 165 = 1*42 + 2*48 +1*27. The Riordan A-sequence is [1,2,1].
%e Recurrence from Riordan Z-sequence [2,1]: T(5,0) = 132 = 2*42 + 1*48. (End)
%e From _Wolfdieter Lang_, Sep 20 2013: (Start)
%e Example for rho(N) = 2*cos(Pi/N) powers:
%e n=2: rho(N)^5 = 5*R(N, 2) + 4*R(N, 4) + 1*R(N, 6) = 5*S(1, rho(N)) + 4*S(3, rho(N)) + 1*S(5, rho(N)), identical in N >= 1. For N=5 (the pentagon with only one distinct diagonal) the degree delta(5) = 2, hence R(5, 4) and R(5, 6) can be reduced, namely to R(5, 1) = 1 and R(5, 6) = -R(5,1) = -1, respectively. Thus rho(5)^5 = 5*R(N, 2) + 4*1 + 1*(-1) = 3 + 5*R(N, 2) = 3 + 5*rho(5), with the golden section rho(5). (End)
%p T:=(n,k)->binomial(2*n, n-k) - binomial(2*n, n-k-2); # _N. J. A. Sloane_, Aug 26 2013
%p # Uses function PMatrix from A357368. Adds row and column above and to the left.
%p PMatrix(10, n -> binomial(2*n, n) / (n + 1)); # _Peter Luschny_, Oct 07 2022
%t Flatten[Table[Binomial[2n, n-k] - Binomial[2n, n-k-2], {n,0,9}, {k,0,n}]] (* _Jean-François Alcover_, May 03 2011 *)
%o (Sage) # Algorithm of L. Seidel (1877)
%o # Prints the first n rows of the triangle.
%o def A039598_triangle(n) :
%o D = [0]*(n+2); D[1] = 1
%o b = True; h = 1
%o for i in range(2*n) :
%o if b :
%o for k in range(h,0,-1) : D[k] += D[k-1]
%o h += 1
%o else :
%o for k in range(1,h, 1) : D[k] += D[k+1]
%o b = not b
%o if b : print([D[z] for z in (1..h-1) ])
%o A039598_triangle(10) # _Peter Luschny_, May 01 2012
%o (Magma) /* As triangle: */ [[Binomial(2*n,n-k) - Binomial(2*n,n-k-2): k in [0..n]]: n in [0.. 15]]; // _Vincenzo Librandi_, Jul 22 2015
%o (PARI) T(n,k)=binomial(2*n,n-k) - binomial(2*n,n-k-2) \\ _Charles R Greathouse IV_, Nov 07 2016
%Y Mirror image of A050166. Row sums are A001700.
%Y Cf. A000108, A008313, A039599, A183134, A094527, A033184, A035324, A053122, A172417.
%K nonn,tabl,easy,nice
%O 0,2
%A _N. J. A. Sloane_
%E Typo in one entry corrected by _Philippe Deléham_, Dec 16 2007
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