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Coefficients of Chebyshev polynomials.
(Formerly M2059)
7

%I M2059 #69 Feb 17 2023 08:15:26

%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 712530,878787,1076103,1308944,1582240,1901416,2272424,2701776,3196578,3764565,4414137,5154396

%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="https://pmf.unibl.org/wp-content/uploads/2017/10/enumfor.pdf">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 Richard K. Guy, <a href="/A005581/a005581_1.pdf">Letter to N. J. A. Sloane, Feb 1988</a>.

%H Simon Plouffe, <a href="https://arxiv.org/abs/0911.4975">Approximations de séries génératrices et quelques conjectures</a>, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.

%H Simon Plouffe, <a href="/A000051/a000051_2.pdf">1031 Generating Functions</a>, Appendix to Thesis, Montreal, 1992.

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

%H Cecilia Rossiter, <a href="/A101096/a101096.pdf">Depictions, Explorations and Formulas of the Euler/Pascal Cube</a>. [Cached copy, May 15 2013]

%H <a href="/index/Rec#order_07">Index entries for linear recurrences with constant coefficients</a>, signature (7,-21,35,-35,21,-7,1).

%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 a(n) = 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

%F From _G. C. Greubel_, Aug 27 2019: (Start)

%F a(n) = (n+11)*Pochhammer(n, 5)/6!.

%F E.g.f.: x*(1440 +3240*x +1920*x^2 +420*x^3 +36*x^4 +x^5)*exp(x)/6!. (End)

%F From _Amiram Eldar_, Feb 17 2023: (Start)

%F Sum_{n>=1} 1/a(n) = 1303391/2134440.

%F Sum_{n>=1} (-1)^(n+1)/a(n) = 4160*log(2)/77 - 78994697/2134440. (End)

%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[2*Binomial[n+4, 5] + Binomial[n+4, 6], {n,40}] (* _Vladimir Joseph Stephan Orlovsky_, Jun 14 2011, modified by _G. C. Greubel_, Aug 27 2019 *)

%t Table[(n+11)*Pochhammer[n, 5]/6!, {n,40}] (* _G. C. Greubel_, Aug 27 2019 *)

%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

%o (Sage) [(n+11)*rising_factorial(n,5)/factorial(6) for n in (1..40)] # _G. C. Greubel_, Aug 27 2019

%o (GAP) List([1..40], n-> (n+11)*Binomial(n+4,5)/6); # _G. C. Greubel_, Aug 27 2019

%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