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 A005584 Coefficients of Chebyshev polynomials. (Formerly M2059) 7

%I M2059

%S 2,13,49,140,336,714,1386,2508,4290,7007,11011,16744,24752,35700,

%T 50388,69768,94962,127281,168245,219604,283360,361790,457470,573300,

%U 712530,878787,1076103,1308944,1582240,1901416

%N Coefficients of Chebyshev polynomials.

%C If X is an n-set and Y a fixed 2-subset of X then a(n-6) is equal to the number of (n-6)-subsets of X intersecting Y. - _Milan Janjic_, Jul 30 2007

%C a(n-1) = risefac(n+1,6)/6! - risefac(n+1,4)/4! is for n >=1 also the number of independent components of a symmetric traceless tensor of rank 6 and dimension n. Here risefac is the rising factorial. Put a(0) = 0. - _Wolfdieter Lang_, Dec 10 2015

%D M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 797.

%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

%H Vincenzo Librandi, <a href="/A005584/b005584.txt">Table of n, a(n) for n = 1..1000</a>

%H Milan Janjic, <a href="http://www.pmfbl.org/janjic/">Two Enumerative Functions</a>

%H 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 R. K. Guy, <a href="/A005581/a005581_1.pdf">Letter to N. J. A. Sloane, Feb 1988</a>

%H Simon Plouffe, <a href="http://www.lacim.uqam.ca/%7Eplouffe/articles/MasterThesis.pdf">Approximations de séries génératrices et quelques conjectures</a>, Dissertation, Université du Québec à Montréal, 1992.

%H Simon Plouffe, <a href="http://www.lacim.uqam.ca/%7Eplouffe/articles/FonctionsGeneratrices.pdf">1031 Generating Functions and Conjectures</a>, Université du Québec à Montréal, 1992.

%H C. Rossiter, <a href="http://noticingnumbers.net/300SeriesCube.htm">Depictions, Explorations and Formulas of the Euler/Pascal Cube</a>.

%H <a href="/index/Ch#Cheby">Index entries for sequences related to Chebyshev polynomials.</a>

%F G.f.: x*(2-x) / (1-x)^7.

%F a(n) = binomial(n+5, n-1) + binomial(n+4, n-1) = 1/720*n*(n+11)*(n+4)*(n+3)*(n+2)*(n+1).

%F Binomial(n,6)+2*binomial(n,5), n >= 5. - _Zerinvary Lajos_, Jul 26 2006

%F a(n+1) = A127672(12+n, n), n >= 0, where A127672 gives the coefficients of Chebyshev's C polynomials. See the Abramowitz-Stegun reference. - _Wolfdieter Lang_, Dec 10 2015

%p [seq(binomial(n,6)+2*binomial(n,5), n=5..35)]; # _Zerinvary Lajos_, Jul 26 2006

%p A005584:=(-2+z)/(z-1)**7; # _Simon Plouffe_ in his 1992 dissertation

%t Table[n (n + 1) (n + 2) (n + 3) (n + 4)/5!, {n, 1, 60}] + Table[n (n + 1) (n + 2) (n + 3) (n + 4) (n + 5)/6!, {n, 1, 60}] (* _Vladimir Joseph Stephan Orlovsky_, Jun 14 2011 *)

%o (PARI) a(n)=n*(n+11)*(n+4)*(n+3)*(n+2)*(n+1)/720 \\ _Charles R Greathouse IV_, Jun 14 2011

%o (MAGMA) [n*(n+11)*(n+4)*(n+3)*(n+2)*(n+1)/720: n in [1..40]]; // _Vincenzo Librandi_, Jun 15 2011

%Y Cf. A127672, A005581, A005582, A005583.

%K nonn,easy

%O 1,1

%A _N. J. A. Sloane_

%E More terms from Klaus Strassburger (strass(AT)ddfi.uni-duesseldorf.de), Dec 07 1999

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Last modified March 26 14:51 EDT 2019. Contains 321497 sequences. (Running on oeis4.)