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A008313 Triangle of expansions of powers of x in terms of Chebyshev polynomials U_n(x). 11

%I

%S 1,1,1,1,2,1,2,3,1,5,4,1,5,9,5,1,14,14,6,1,14,28,20,7,1,42,48,27,8,1,

%T 42,90,75,35,9,1,132,165,110,44,10,1,132,297,275,154,54,11,1,429,572,

%U 429,208,65,12,1,429,1001,1001,637,273,77,13,1,1430,2002,1638,910,350

%N Triangle of expansions of powers of x in terms of Chebyshev polynomials U_n(x).

%C This is another reading (by shallow diagonals) of the triangle A009766; rows of Catalan triangle A008315 read backwards. - _Philippe Deléham_, Feb 15 2004

%C "The Catalan triangle is formed in the same manner as Pascal's triangle, except that no number may appear on the left of the vertical bar." [Conway and Smith]

%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 J. H. Conway and D. A. Smith, On Quaternions and Octonions, A K Peters, Ltd., Natick, MA, 2003. See p. 60. MR1957212 (2004a:17002)

%D P. J. Larcombe, A question of proof..., Bull. Inst. Math. Applic. (IMA), 30, Nos. 3/4, 1994, 52-54.

%H Reinhard Zumkeller, <a href="/A008313/b008313.txt">Rows n=0..150 of triangle, flattened</a>

%H M. Abramowitz and I. A. Stegun, eds., <a href="http://apps.nrbook.com/abramowitz_and_stegun/index.html">Handbook of Mathematical Functions</a>, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].

%H I. Dolinka, J. East, A. Evangelou, D. FitzGerald, N. Ham, <a href="http://arxiv.org/abs/1507.04838">Idempotent Statistics of the Motzkin and Jones Monoids</a>, arXiv preprint arXiv:1507.04838, 2015

%H Vaughan F. R. Jones, <a href="http://www.math.berkeley.edu/~vfr/jones.pdf">The Jones Polynomial</a>, 18 August 2005, see the diagram on page 7. - Paul Curtz, Jun 22 2011

%H P. Mongelli, <a href="http://arxiv.org/abs/1111.2945">Kazhdan-Lusztig polynomials of Boolean elements</a>, arXiv preprint arXiv:1111.2945, 2011

%H <a href="/index/Ch#Cheby">Index entries for sequences related to Chebyshev polynomials.</a>

%F Row n: C(n-1, [ n/2 ]-k)-C(n-1, [ n/2 ]-k-2), k=0, 1, ..., n.

%F Sum_{k>=0} T(n, k)^2 = A000108(n); A000108: Catalan numbers. - _Philippe Deléham_, Feb 14 2004

%e .|...1

%e .|.......1

%e .|...1.......1

%e .|.......2.......1

%e .|...2.......3.......1

%e .|.......5.......4.......1

%e .|...5.......9.......5.......1

%e .|......14......14.......6.......1

%e .|..14......28......20.......7.......1

%e .|......42......48......27.......8.......1

%p T := proc(n, k): if n=0 then 1 else binomial(n-1, floor(n/2 )-k) -binomial(n-1, floor(n/2) -k-2) fi: end: seq(seq(T(n, k), k = 0..floor(n/2)), n = 0..14); # _Johannes W. Meijer_, Jul 10 2011, revised Nov 22 2012

%t t[n_, k_] /; n < k || OddQ[n - k] = 0; t[n_, k_] := (k+1)*Binomial[n+1, (n-k)/2]/(n+1); Flatten[ Table[ t[n, k], {n, 0, 15}, {k, Mod[n, 2], n + Mod[n, 2], 2}]] (* _Jean-François Alcover_, Jan 12 2012 *)

%o (PARI) {T(n, k) = if( k<0 || 2*k>n, 0, polcoeff((1 - x) * (1 + x)^n, n\2 - k))}; /* _Michael Somos_, May 28 2005 */

%o (Haskell)

%o a008313 n k = a008313_tabf !! n !! k

%o a008313_row n = a008313_tabf !! n

%o a008313_tabf = map (filter (> 0)) a053121_tabl

%o -- _Reinhard Zumkeller_, Feb 24 2012

%o (Sage) # Algorithm of L. Seidel (1877)

%o # Prints the first n rows of the triangle.

%o def A008313_triangle(n) :

%o D = [0]*((n+5)//2); D[1] = 1

%o b = True; h = 1

%o for i in range(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 print [D[z] for z in (1..h-1)]

%o A008313_triangle(13) # _Peter Luschny_, May 01 2012

%Y Cf. A039598, A039599. A053121 is essentially the same triangle.

%Y Row sums = A001405 (central binomial coefficients).

%K nonn,tabf,nice,easy

%O 0,5

%A _N. J. A. Sloane_

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Last modified February 25 22:27 EST 2018. Contains 299662 sequences. (Running on oeis4.)