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A008313
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Triangle of expansions of powers of x in terms of Chebyshev polynomials U_n(x).
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11
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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
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OFFSET
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0,5
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COMMENTS
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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]
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REFERENCES
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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.
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LINKS
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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].
Vaughan F. R. Jones, The Jones Polynomial, 18 August 2005, see the diagram on page 7. - Paul Curtz, Jun 22 2011
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FORMULA
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Row n: C(n-1, [ n/2 ]-k)-C(n-1, [ n/2 ]-k-2), k=0, 1, ..., n.
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EXAMPLE
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.|...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
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MAPLE
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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
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MATHEMATICA
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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 *)
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PROG
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(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
(Sage) # Algorithm of L. Seidel (1877)
# Prints the first n rows of the triangle.
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)])
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CROSSREFS
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Row sums = A001405 (central binomial coefficients).
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KEYWORD
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nonn,tabf,nice,easy
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AUTHOR
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STATUS
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approved
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