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A005583 Coefficients of Chebyshev polynomials.

%I A005583 M1999 #92 Feb 17 2023 10:04:04
%S A005583 2,11,36,91,196,378,672,1122,1782,2717,4004,5733,8008,10948,14688,
%T A005583 19380,25194,32319,40964,51359,63756,78430,95680,115830,139230,166257,
%U A005583 197316,232841,273296,319176,371008,429352,494802,567987,649572,740259,840788
%N A005583 Coefficients of Chebyshev polynomials.
%C A005583 If X is an n-set and Y a fixed 2-subset of X then a(n-5) is equal to the number of (n-5)-subsets of X intersecting Y. - _Milan Janjic_, Jul 30 2007
%C A005583 a(n-1) = risefac(n,5)/5! - risefac(n,3)/3! is for n >= 1 also the number of independent components of a symmetric traceless tensor of rank 5 and dimension n. Here risefac is the rising factorial. Put a(0) = 0. - _Wolfdieter Lang_, Dec 10 2015
%D A005583 M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), Table 22.7, p. 797.
%D A005583 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H A005583 Vincenzo Librandi, <a href="/A005583/b005583.txt">Table of n, a(n) for n = 1..172</a>
%H A005583 Milan Janjic, <a href="https://pmf.unibl.org/wp-content/uploads/2017/10/enumfor.pdf">Two Enumerative Functions</a>.
%H A005583 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 A005583 Richard K. Guy, <a href="/A005581/a005581_1.pdf">Letter to N. J. A. Sloane, Feb 1988</a>.
%H A005583 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 A005583 Simon Plouffe, <a href="/A000051/a000051_2.pdf">1031 Generating Functions</a>, Appendix to Thesis, Montreal, 1992.
%H A005583 Cecilia Rossiter, <a href="http://noticingnumbers.net/300SeriesCube.htm">Depictions, Explorations and Formulas of the Euler/Pascal Cube</a>. [broken link]
%H A005583 Cecilia Rossiter, <a href="/A101096/a101096.pdf">Depictions, Explorations and Formulas of the Euler/Pascal Cube</a>. [Cached copy, May 15 2013]
%H A005583 <a href="/index/Ch#Cheby">Index entries for sequences related to Chebyshev polynomials</a>.
%F A005583 G.f.: x*(2-x)/(1-x)^6.
%F A005583 a(n) = binomial(n+4, n-1) + binomial(n+3, n-1) = (1/120)*n*(n+9)*(n+3)*(n+2)*(n+1).
%F A005583 a(n+1) = -A127672(10+n, n), n >= 0, with the coefficients of the Chebyshev C-polynomials A127672(n, k). - _Wolfdieter Lang_, Dec 10 2015
%F A005583 a(n) = Sum_{i=1..n} A000217(i)*A000096(n+1-i). - _Bruno Berselli_, Mar 05 2018
%F A005583 a(n) = binomial(n+3,5) + 2*binomial(n+3,4). - _Yuchun Ji_, May 23 2019
%F A005583 From _Amiram Eldar_, Feb 17 2023: (Start)
%F A005583 Sum_{n>=1} 1/a(n) = 40751/63504.
%F A005583 Sum_{n>=1} (-1)^(n+1)/a(n) = 1360*log(2)/63 - 922961/63504. (End)
%p A005583 A005583:=-(-2+z)/(z-1)**6; # _Simon Plouffe_ in his 1992 dissertation (this g.f. assumes offset 0)
%o A005583 (PARI)
%o A005583 conv(u,v)=local(w); w=vector(length(u),i,sum(j=1,i,u[j]*v[i+1-j])); w;
%o A005583 t(n)=n*(n+1)/2;
%o A005583 u=vector(10,i,t(i));
%o A005583 v=vector(10,i,t(i)-1);
%o A005583 conv(u,v)
%o A005583 (PARI) a(n) = (1/120)*n*(n+9)*(n+3)*(n+2)*(n+1); \\ _Joerg Arndt_, Mar 05 2018
%Y A005583 Cf. A000096, A000217, A000389, A051747, A127672.
%Y A005583 Column 3 of A207606.
%K A005583 nonn,easy
%O A005583 1,1
%A A005583 _N. J. A. Sloane_
%E A005583 More terms from Klaus Strassburger (strass(AT)ddfi.uni-duesseldorf.de), Dec 07 1999
%E A005583 More terms from _Zerinvary Lajos_, Jul 21 2006