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 A089627 T(n,k) = binomial(n,2*k)*binomial(2*k,k) for 0 <= k <= n, triangle read by rows. 7
 1, 1, 0, 1, 2, 0, 1, 6, 0, 0, 1, 12, 6, 0, 0, 1, 20, 30, 0, 0, 0, 1, 30, 90, 20, 0, 0, 0, 1, 42, 210, 140, 0, 0, 0, 0, 1, 56, 420, 560, 70, 0, 0, 0, 0, 1, 72, 756, 1680, 630, 0, 0, 0, 0, 0, 1, 90, 1260, 4200, 3150, 252, 0, 0, 0, 0, 0, 1, 110, 1980, 9240, 11550, 2772, 0, 0, 0, 0, 0, 0, 1, 132, 2970, 18480, 34650, 16632, 924, 0, 0, 0, 0, 0, 0, 1, 156, 4290, 34320, 90090, 72072, 12012, 0, 0, 0, 0, 0, 0, 0, 1, 182, 6006, 60060, 210210, 252252, 84084, 3432, 0, 0, 0, 0, 0, 0, 0 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 0,5 COMMENTS The rows of this triangle are the gamma vectors of the n-dimensional type B associahedra (Postnikov et al., p.38 ). Cf. A055151 and A101280. - Peter Bala, Oct 28 2008 T(n,k) is the number of Grand Motzkin paths of length n having exactly k upsteps (1,1). Cf. A109189, A055151. - Geoffrey Critzer, Feb 05 2014 The result Sum_{k = 0..floor(n/2)} C(n,2*k)*C(2*k,k)*x^k = (sqrt(1 - 4*x))^n* P(n,1/sqrt(1 - 4*x)) expressing the row polynomials of this triangle in terms of the Legendre polynomials P(n,x) is due to Catalan. See Laden, equation 7.10, p. 56. - Peter Bala, Mar 18 2018 LINKS Alois P. Heinz, Rows n = 0..140, flattened Rui Duarte and António Guedes de Oliveira, A Famous Identity of Hajós in Terms of Sets, Journal of Integer Sequences, Vol. 17 (2014), #14.9.1. Hyman N. Laden, An historical, and critical development of the theory of Legendre polynomials before 1900, Master of Arts Thesis, University of Maryland 1938. Shi-Mei Ma, On gamma-vectors and the derivatives of the tangent and secant functions, arXiv:1304.6654 [math.CO], 2013. A. Postnikov, V. Reiner, and L. Williams, Faces of generalized permutohedra, arXiv:math/0609184 [math.CO], 2006-2007. FORMULA T(n,k) = n!/((n-2*k)!*k!*k!). E.g.f.: exp(x)*BesselI(0, 2*x*sqrt(y)). - Vladeta Jovovic, Apr 07 2005 O.g.f.: ( 1 - x - sqrt(1 - 2*x + x^2 - 4*x^2*y))/(2*x^2*y). - Geoffrey Critzer, Feb 05 2014 R(n, x) = hypergeom([1/2 - n/2, -n/2], [1], 4*x) are the row polynomials. - Peter Luschny, Mar 18 2018 From Peter Bala, Jun 23 2023: (Start) T(n,k) = Sum_{i = 0..k} (-1)^i*binomial(n, i)*binomial(n-i, k-i)^2. Cf. A063007(n,k) = Sum_{i = 0..k} binomial(n, i)^2*binomial(n-i, k-i). T(n,k) = A063007(n-k,k); that is, the diagonals of this table are the rows of A063007. (End) EXAMPLE Triangle begins: 1 1, 0 1, 2, 0 1, 6, 0, 0 1, 12, 6, 0, 0 1, 20, 30, 0, 0, 0 1, 30, 90, 20, 0, 0, 0 1, 42, 210, 140, 0, 0, 0, 0 1, 56, 420, 560, 70, 0, 0, 0, 0 1, 72, 756, 1680, 630, 0, 0, 0, 0, 0 1, 90, 1260, 4200, 3150, 252, 0, 0, 0, 0, 0 1, 110, 1980, 9240, 11550, 2772, 0, 0, 0, 0, 0, 0 1, 132, 2970, 18480, 34650, 16632, 924, 0, 0, 0, 0, 0, 0 Relocating the zeros to be evenly distributed and interpreting the triangle as the coefficients of polynomials 1 1 1 + 2 q^2 1 + 6 q^2 1 + 12 q^2 + 6 q^4 1 + 20 q^2 + 30 q^4 1 + 30 q^2 + 90 q^4 + 20 q^6 1 + 42 q^2 + 210 q^4 + 140 q^6 1 + 56 q^2 + 420 q^4 + 560 q^6 + 70 q^8 then the substitution q^k -> 1/(floor(k/2)+1) gives the Motzkin numbers A001006. - Peter Luschny, Aug 29 2011 MAPLE for i from 0 to 12 do seq(binomial(i, j)*binomial(i-j, j), j=0..i) od; # Zerinvary Lajos, Jun 07 2006 # Alternatively: R := (n, x) -> simplify(hypergeom([1/2 - n/2, -n/2], [1], 4*x)): Trow := n -> seq(coeff(R(n, x), x, j), j=0..n): seq(print(Trow(n)), n=0..9); # Peter Luschny, Mar 18 2018 MATHEMATICA nn=15; mxy=(1-x-(1-2x+x^2-4x^2y)^(1/2))/(2x^2 y); Map[Select[#, #>0&]&, CoefficientList[Series[1/(1-x-2y x^2mxy), {x, 0, nn}], {x, y}]]//Grid (* Geoffrey Critzer, Feb 05 2014 *) PROG (PARI) T(n, k) = binomial(n, 2*k)*binomial(2*k, k); concat(vector(15, n, vector(n, k, T(n-1, k-1)))) \\ Gheorghe Coserea, Sep 01 2018 CROSSREFS Row sums A002426. Antidiagonal sums A098479. Cf. A055151, A063007, A109187, A101280. Sequence in context: A151860 A338774 A330891 * A306534 A344392 A331787 Adjacent sequences: A089624 A089625 A089626 * A089628 A089629 A089630 KEYWORD easy,nonn,tabl AUTHOR Philippe Deléham, Dec 31 2003 STATUS approved

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Last modified August 11 21:32 EDT 2024. Contains 375073 sequences. (Running on oeis4.)