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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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7
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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
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,5
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COMMENTS
| This is another reading (by shallow diagonals) of the triangle A009766; rows of Catalan triangle A008315 read backwards. - DELEHAM Philippe (kolotoko(AT)wanadoo.fr), 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
| 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.
P. Mongelli, Kazhdan-Lusztig polynomials of Boolean elements, Arxiv preprint arXiv:1111.2945, 2011
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LINKS
| 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, June 22 2011
Index entries for sequences related to Chebyshev polynomials.
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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: numbers of Catalan . - DELEHAM Philippe (kolotoko(AT)wanadoo.fr), Feb 14 2004
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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
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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: for n from 0 to 10 do seq(T(n, k), k=0..floor(n/2)) od: nmax:=14; Tx:=0: for n from 0 to nmax do for k from 0 to floor(n/2) do a(Tx):=T(n, k): Tx:=Tx+1: od: od: seq(a(n), n=0..Tx-1); [Johannes W. Meijer, Jul 10 2011]
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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}]] (* From Jean-François Alcover, Jan 12 2012 *)
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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 */
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CROSSREFS
| Cf. A039598, A039599. A053121 is essentially the same triangle.
Row sums = A001405 (central binomial coefficients).
Sequence in context: A117704 A078032 A162453 * A111377 A014046 A128065
Adjacent sequences: A008310 A008311 A008312 * A008314 A008315 A008316
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KEYWORD
| nonn,tabf,nice,easy
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com).
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EXTENSIONS
| More terms from Clark Kimberling (ck6(AT)evansville.edu)
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