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

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• tabf
• nice
• easy

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]

Examples

			.|...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
		

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.

Crossrefs

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

Row sums = A001405 (central binomial coefficients).

Programs

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

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 *)

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 */

PARI

T(n, k) = binomial(n-1, n\2-k)-binomial(n-1, n\2-k-2);
for(n=0, 14, for(k=0, n\2, print1(T(n,k),", "))); \\ Seiichi Manyama, Mar 24 2025

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

Formula

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

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