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

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, 42, 90, 75, 35, 9, 1, 132, 165, 110, 44, 10, 1, 132, 297, 275, 154, 54, 11, 1, 429, 572, 429, 208, 65, 12, 1, 429, 1001, 1001, 637, 273, 77, 13, 1, 1430, 2002, 1638, 910, 350

Offset

0,5

Comments

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

"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]

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.

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

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

Links

Reinhard Zumkeller, Rows n=0..150 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].

I. Dolinka, J. East, A. Evangelou, D. FitzGerald, N. Ham, Idempotent Statistics of the Motzkin and Jones Monoids, arXiv preprint arXiv:1507.04838 [math.CO], 2015-2018.

Tom Halverson, Theodore N. Jacobson, Set-partition tableaux and representations of diagram algebras, arXiv:1808.08118 [math.RT], 2018.

Vaughan F. R. Jones, The Jones Polynomial, 18 August 2005, see the diagram on page 7. - Paul Curtz, Jun 22 2011

P. Mongelli, Kazhdan-Lusztig polynomials of Boolean elements, arXiv preprint arXiv:1111.2945 [math.CO], 2011.

Index entries for sequences related to Chebyshev polynomials.

Formula

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

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

Example

.|...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

Maple

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

Mathematica

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 *)

Prog

(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 */
(Haskell)
a008313 n k = a008313_tabf !! n !! k
a008313_row n = a008313_tabf !! n
a008313_tabf = map (filter (> 0)) a053121_tabl
-- Reinhard Zumkeller, Feb 24 2012
(Sage) # Algorithm of L. Seidel (1877)
# Prints the first n rows of the triangle.
def A008313_triangle(n) :
    D = [0]*((n+5)//2); D[1] = 1
    b = True; h = 1
    for i in range(n) :
        if b :
            for k in range(h, 0, -1) : D[k] += D[k-1]
            h += 1
        else :
            for k in range(1, h, 1) : D[k] += D[k+1]
        b = not b
        print([D[z] for z in (1..h-1)])
A008313_triangle(13) # Peter Luschny, May 01 2012

Crossrefs

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

Row sums = A001405 (central binomial coefficients).

Sequence in context: A117704 A078032 A162453 * A334550 A232177 A111377

Adjacent sequences: A008310 A008311 A008312 * A008314 A008315 A008316

Keyword

nonn,tabf,nice,easy

Author

N. J. A. Sloane

Status

approved