A005583 Coefficients of Chebyshev polynomials.

2, 11, 36, 91, 196, 378, 672, 1122, 1782, 2717, 4004, 5733, 8008, 10948, 14688, 19380, 25194, 32319, 40964, 51359, 63756, 78430, 95680, 115830, 139230, 166257, 197316, 232841, 273296, 319176, 371008, 429352, 494802, 567987, 649572, 740259, 840788

Offset: 1

Author

N. J. A. Sloane

Keywords

• nonn
• easy

Comments

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

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

References

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

Links

• Vincenzo Librandi, Table of n, a(n) for n = 1..172
• Milan Janjic, Two Enumerative Functions.
• 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].
• Richard K. Guy, Letter to N. J. A. Sloane, Feb 1988.
• Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
• Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992.
• Cecilia Rossiter, Depictions, Explorations and Formulas of the Euler/Pascal Cube. [broken link]
• Cecilia Rossiter, Depictions, Explorations and Formulas of the Euler/Pascal Cube. [Cached copy, May 15 2013]
• Index entries for sequences related to Chebyshev polynomials.

Crossrefs

Cf. A000096, A000217, A000389, A051747, A127672.

Column 3 of A207606.

Programs

Maple

A005583:=-(-2+z)/(z-1)**6; # Simon Plouffe in his 1992 dissertation (this g.f. assumes offset 0)

PARI

conv(u,v)=local(w); w=vector(length(u),i,sum(j=1,i,u[j]*v[i+1-j])); w;
t(n)=n*(n+1)/2;
u=vector(10,i,t(i));
v=vector(10,i,t(i)-1);
conv(u,v)

PARI

a(n) = (1/120)*n*(n+9)*(n+3)*(n+2)*(n+1); \\ Joerg Arndt, Mar 05 2018

Formula

G.f.: x*(2-x)/(1-x)^6.

a(n) = binomial(n+4, n-1) + binomial(n+3, n-1) = (1/120)*n*(n+9)*(n+3)*(n+2)*(n+1).

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

a(n) = Sum_{i=1..n} A000217(i)*A000096(n+1-i). - Bruno Berselli, Mar 05 2018

a(n) = binomial(n+3,5) + 2*binomial(n+3,4). - Yuchun Ji, May 23 2019

From Amiram Eldar, Feb 17 2023: (Start)

Sum_{n>=1} 1/a(n) = 40751/63504.

Sum_{n>=1} (-1)^(n+1)/a(n) = 1360*log(2)/63 - 922961/63504. (End)

Extensions

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

More terms from Zerinvary Lajos, Jul 21 2006