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 A008315 Catalan triangle read by rows. Also triangle of expansions of powers of x in terms of Chebyshev polynomials U_n(x). 32
 1, 1, 1, 1, 1, 2, 1, 3, 2, 1, 4, 5, 1, 5, 9, 5, 1, 6, 14, 14, 1, 7, 20, 28, 14, 1, 8, 27, 48, 42, 1, 9, 35, 75, 90, 42, 1, 10, 44, 110, 165, 132, 1, 11, 54, 154, 275, 297, 132, 1, 12, 65, 208, 429, 572, 429, 1, 13, 77, 273, 637, 1001, 1001, 429, 1, 14, 90, 350, 910, 1638, 2002, 1430, 1, 15, 104 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,6 COMMENTS There are several versions of a Catalan triangle: see A009766, A008315, A028364, A053121. Number of standard tableaux of shape (n-k,k) (0<=k<=floor(n/2)). Example: T(4,1)=3 because in th top row we can have 124, 134, or 123 (but not 234). - Emeric Deutsch, May 23 2004 T(n,k) is the number of n-digit binary words (length n sequences on {0,1}) containing k 1's such that no initial segment of the sequence has more 1's than 0's. - Geoffrey Critzer, Jul 31 2009 T(n,k) is the number of dispersed Dyck paths (i.e. Motzkin paths with no (1,0) steps at positive heights) of length n and having k (1,1)-steps. Example: T(5,1)=4 because, denoting U=(1,1), D=(1,-1), H=1,0), we have HHHUD, HHUDH, HUDHH, and UDHHH. - Emeric Deutsch, May 30 2011 T(n,k) is the number of length n left factors of Dyck paths having k (1,-1)-steps. Example: T(5,1)=4 because, denoting U=(1,1), D=(1,-1), we have UUUUD, UUUDU, UUDUU, and UDUUU. There is a simple bijection between length n left factors of Dyck paths and dispersed Dyck paths of length n, that takes D steps into D steps. - Emeric Deutsch, Jun 19 2011 Triangle, with zeros omitted, given by (1, 0, 0, -1, 1, 0, 0, -1, 1, 0, 0, -1, 1, ...) DELTA (0, 1, -1, 0, 0, 1, -1, 0, 0, 1, -1, 0, 0, 1, ...) where DELTA is the operator defined in A084938. - Philippe Deléham, Dec 12 2011 T(n,k) are rational multiples of A055151(n,k). - Peter Luschny, Oct 16 2015 T(2*n,n) = Sum_{k>=0} T(n,k)^2 = A000108(n), T(2*n+1,n) = A000108(n+1). - Michael Somos, Jun 08 2020 REFERENCES 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. P. J. Larcombe, A question of proof..., Bull. Inst. Math. Applic. (IMA), 30, Nos. 3/4, 1994, 52-54. LINKS T. D. Noe, Rows n=0..100 of triangle, flattened M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy]. Tewodros Amdeberhan, Moa Apagodu, and Doron Zeilberger, Wilf's "Snake Oil" Method Proves an Identity in The Motzkin Triangle, arXiv:1507.07660 [math.CO], 2015. Nantel Bergeron, Kelvin Chan, Yohana Solomon, Farhad Soltani, and Mike Zabrocki, Quasisymmetric harmonics of the exterior algebra, arXiv:2206.02065 [math.CO], 2022. Suyoung Choi and Hanchul Park, A new graph invariant arises in toric topology, arXiv preprint arXiv:1210.3776 [math.AT], 2012. C. Kenneth Fan, Structure of a Hecke algebra quotient, J. Amer. Math. Soc. 10 (1997), no. 1, 139-167. R. K. Guy, Catwalks, sandsteps and Pascal pyramids, J. Integer Sequences, Vol. 3 (2000), Article #00.1.6. L. Jiu, V. H. Moll, and C. Vignat, Identities for generalized Euler polynomials, arXiv:1401.8037 [math.PR], 2014. N. Lygeros and O. Rozier, A new solution to the equation tau(rho) == 0 (mod p), J. Int. Seq. 13 (2010) # 10.7.4. M. A. A. Obaid, S. K. Nauman, W. M. Fakieh, and C. M. Ringel, The numbers of support-tilting modules for a Dynkin algebra, 2014 and J. Int. Seq. 18 (2015) 15.10.6. Alon Regev, The central component of a triangulation, Journal of Integer Sequences, Vol. 16 (2013), Article 13.4.1, p. 7. J. Riordan, The distribution of crossings of chords joining pairs of 2n points on a circle, Math. Comp., 29 (1975), 215-222. J. Riordan, The distribution of crossings of chords joining pairs of 2n points on a circle, Math. Comp., 29 (1975), 215-222. [Annotated scanned copy] L. W. Shapiro, A Catalan triangle, Discrete Math. 14 (1976), no. 1, 83-90. [Annotated scanned copy] Zheng Shi, Impurity entropy of junctions of multiple quantum wires, arXiv preprint arXiv:1602.00068 [cond-mat.str-el], 2016 (See Appendix A). Index entries for sequences related to Chebyshev polynomials. FORMULA T(n, 0) = 1 if n >= 0; T(2*k, k) = T(2*k-1, k-1) if k>0; T(n, k) = T(n-1, k-1) + T(n-1, k) if k=1, 2, ..., floor(n/2). - Michael Somos, Aug 17 1999 T(n, k) = binomial(n, k) - binomial(n, k-1). - Michael Somos, Aug 17 1999 Rows of Catalan triangle A008313 read backwards. Sum_{k>=0} T(n, k)^2 = A000108(n); A000108 : Catalan numbers. - Philippe Deléham, Feb 15 2004 T(n,k) = C(n,k)*(n-2*k+1)/(n-k+1). - Geoffrey Critzer, Jul 31 2009 Sum_{k=0..n} T(n,k)*x^k = A000012(n), A001405(n), A126087(n), A128386(n), A121724(n), A128387(n), A132373(n), A132374(n), A132375(n), A121725(n) for x = 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 respectively. - Philippe Deléham, Dec 12 2011 EXAMPLE Triangle begins: 1; 1; 1, 1; 1, 2; 1, 3, 2; 1, 4, 5; 1, 5, 9, 5; 1, 6, 14, 14; 1, 7, 20, 28, 14; ... T(5,2) = 5 because there are 5 such sequences: {0, 0, 0, 1, 1}, {0, 0, 1, 0, 1}, {0, 0, 1, 1, 0}, {0, 1, 0, 0, 1}, {0, 1, 0, 1, 0}. - Geoffrey Critzer, Jul 31 2009 MAPLE b:= proc(x, y) option remember; `if`(y<0 or y>x, 0, `if`(x=0, 1, add(b(x-1, y+j), j=[-1, 1]))) end: T:= (n, k)-> b(n, n-2*k): seq(seq(T(n, k), k=0..n/2), n=0..16); # Alois P. Heinz, Oct 14 2022 MATHEMATICA Table[Binomial[k, i]*(k - 2 i + 1)/(k - i + 1), {k, 0, 20}, {i, 0, Floor[k/2]}] // Grid (* Geoffrey Critzer, Jul 31 2009 *) PROG (PARI) {T(n, k) = if( k<0 || k>n\2, 0, if( n==0, 1, T(n-1, k-1) + T(n-1, k)))}; /* Michael Somos, Aug 17 1999 */ (Haskell) a008315 n k = a008315_tabf !! n !! k a008315_row n = a008315_tabf !! n a008315_tabf = map reverse a008313_tabf -- Reinhard Zumkeller, Nov 14 2013 CROSSREFS T(2n, n) = A000108 (Catalan numbers), row sums = A001405 (central binomial coefficients). This is also the positive half of the triangle in A008482. - Michael Somos This is another reading (by shallow diagonals) of the triangle A009766, i.e. A008315[n] = A009766[A056536[n]]. Cf. A120730, A055151. Sequence in context: A165999 A049280 A108786 * A191318 A341315 A293600 Adjacent sequences: A008312 A008313 A008314 * A008316 A008317 A008318 KEYWORD nonn,tabf,nice,easy AUTHOR N. J. A. Sloane EXTENSIONS Expanded description from Clark Kimberling, Jun 15 1997 STATUS approved

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Last modified May 28 15:40 EDT 2023. Contains 363019 sequences. (Running on oeis4.)