Search: id:A008313
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%I A008313
%S A008313 1,1,1,1,2,1,2,3,1,5,4,1,5,9,5,1,14,14,6,1,14,28,20,7,1,42,48,27,8,1,42,
%T A008313 90,75,35,9,1,132,165,110,44,10,1,132,297,275,154,54,11,1,429,572,429,
%U A008313 208,65,12,1,429,1001,1001,637,273,77,13,1,1430,2002,1638,910,350
%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. - DELEHAM Philippe 
               (kolotoko(AT)wanadoo.fr), 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 M. Abramowitz and I. A. Stegun, eds., <a href="http://www.nrbook.com/
               abramowitz_and_stegun/">Handbook of Mathematical Functions</a>, National 
               Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 
               [alternative scanned copy].
%H A008313 <a href="Sindx_Ch.html#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), k=0, 1, ..., n.
%F A008313 Sum_{k>=0} T(n, k)^2 = A000108(n); A000108: numbers of Catalan . - DELEHAM 
               Philippe (kolotoko(AT)wanadoo.fr), 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
%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 */
%Y A008313 Cf. A039598, A039599. A053121 is essentially the same triangle.
%Y A008313 Row sums = A001405 (central binomial coefficients).
%Y A008313 Sequence in context: A117704 A078032 A162453 this_sequence A111377 A014046 
               A128065
%Y A008313 Adjacent sequences: A008310 A008311 A008312 this_sequence A008314 A008315 
               A008316
%K A008313 nonn,tabf,nice,easy
%O A008313 0,5
%A A008313 N. J. A. Sloane (njas(AT)research.att.com).
%E A008313 More terms from Clark Kimberling (ck6(AT)evansville.edu)

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