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

%I A008313 #73 Mar 24 2025 10:21:47
%S A008313 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 A008313 42,90,75,35,9,1,132,165,110,44,10,1,132,297,275,154,54,11,1,429,572,
%U A008313 429,208,65,12,1,429,1001,1001,637,273,77,13,1
%N A008313 Triangle of expansions of powers of x in terms of Chebyshev polynomials U_n(x).
%C A008313 This is another reading (by shallow diagonals) of the triangle A009766; rows of Catalan triangle A008315 read backwards. - _Philippe Deléham_, Feb 15 2004
%C A008313 "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 A008313 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 A008313 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 A008313 P. J. Larcombe, A question of proof..., Bull. Inst. Math. Applic. (IMA), 30, Nos. 3/4, 1994, 52-54.
%H A008313 Reinhard Zumkeller, <a href="/A008313/b008313.txt">Rows n=0..150 of triangle, flattened</a>
%H A008313 M. Abramowitz and I. A. Stegun, eds., <a href="http://www.convertit.com/Go/ConvertIt/Reference/AMS55.ASP">Handbook of Mathematical Functions</a>, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
%H A008313 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 [math.CO], 2015-2018.
%H A008313 Tom Halverson, Theodore N. Jacobson, <a href="https://arxiv.org/abs/1808.08118">Set-partition tableaux and representations of diagram algebras</a>, arXiv:1808.08118 [math.RT], 2018.
%H A008313 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 A008313 P. Mongelli, <a href="http://arxiv.org/abs/1111.2945">Kazhdan-Lusztig polynomials of Boolean elements</a>, arXiv preprint arXiv:1111.2945 [math.CO], 2011.
%H A008313 <a href="/index/Ch#Cheby">Index entries for sequences related to Chebyshev polynomials.</a>
%F A008313 Row n: C(n-1, [n/2]-k) - C(n-1, [n/2]-k-2) for k=0, 1, ..., n.
%F A008313 Sum_{k>=0} T(n, k)^2 = A000108(n); A000108: Catalan numbers. - _Philippe Deléham_, Feb 14 2004
%e A008313 .|...1
%e A008313 .|.......1
%e A008313 .|...1.......1
%e A008313 .|.......2.......1
%e A008313 .|...2.......3.......1
%e A008313 .|.......5.......4.......1
%e A008313 .|...5.......9.......5.......1
%e A008313 .|......14......14.......6.......1
%e A008313 .|..14......28......20.......7.......1
%e A008313 .|......42......48......27.......8.......1
%p A008313 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 A008313 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 A008313 (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 A008313 (PARI) T(n, k) = binomial(n-1, n\2-k)-binomial(n-1, n\2-k-2);
%o A008313 for(n=0, 14, for(k=0, n\2, print1(T(n,k),", "))); \\ _Seiichi Manyama_, Mar 24 2025
%o A008313 (Haskell)
%o A008313 a008313 n k = a008313_tabf !! n !! k
%o A008313 a008313_row n = a008313_tabf !! n
%o A008313 a008313_tabf = map (filter (> 0)) a053121_tabl
%o A008313 -- _Reinhard Zumkeller_, Feb 24 2012
%o A008313 (Sage) # Algorithm of L. Seidel (1877)
%o A008313 # Prints the first n rows of the triangle.
%o A008313 def A008313_triangle(n) :
%o A008313     D = [0]*((n+5)//2); D[1] = 1
%o A008313     b = True; h = 1
%o A008313     for i in range(n) :
%o A008313         if b :
%o A008313             for k in range(h,0,-1) : D[k] += D[k-1]
%o A008313             h += 1
%o A008313         else :
%o A008313             for k in range(1,h, 1) : D[k] += D[k+1]
%o A008313         b = not b
%o A008313         print([D[z] for z in (1..h-1)])
%o A008313 A008313_triangle(13) # _Peter Luschny_, May 01 2012
%Y A008313 Cf. A039598, A039599. A053121 is essentially the same triangle.
%Y A008313 Row sums = A001405 (central binomial coefficients).
%K A008313 nonn,tabf,nice,easy
%O A008313 0,5
%A A008313 _N. J. A. Sloane_