A005584 Coefficients of Chebyshev polynomials.

2, 13, 49, 140, 336, 714, 1386, 2508, 4290, 7007, 11011, 16744, 24752, 35700, 50388, 69768, 94962, 127281, 168245, 219604, 283360, 361790, 457470, 573300, 712530, 878787, 1076103, 1308944, 1582240, 1901416, 2272424, 2701776, 3196578, 3764565, 4414137, 5154396

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-6) is equal to the number of (n-6)-subsets of X intersecting Y. - Milan Janjic, Jul 30 2007

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

References

• 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.
• 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..1000
• 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.
• Cecilia Rossiter, Depictions, Explorations and Formulas of the Euler/Pascal Cube. [Cached copy, May 15 2013]
• Index entries for linear recurrences with constant coefficients, signature (7,-21,35,-35,21,-7,1).
• Index entries for sequences related to Chebyshev polynomials.

Crossrefs

Cf. A127672, A005581, A005582, A005583.

Programs

GAP

List([1..40], n-> (n+11)*Binomial(n+4,5)/6); # G. C. Greubel, Aug 27 2019

Magma

[n*(n+11)*(n+4)*(n+3)*(n+2)*(n+1)/720: n in [1..40]]; // Vincenzo Librandi, Jun 15 2011

Maple

[seq(binomial(n,6)+2*binomial(n,5), n=5..35)]; # Zerinvary Lajos, Jul 26 2006
A005584:=(-2+z)/(z-1)**7; # Simon Plouffe in his 1992 dissertation

Mathematica

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 *)
Table[(n+11)*Pochhammer[n, 5]/6!, {n,40}] (* G. C. Greubel, Aug 27 2019 *)

PARI

a(n)=n*(n+11)*(n+4)*(n+3)*(n+2)*(n+1)/720 \\ Charles R Greathouse IV, Jun 14 2011

Sage

[(n+11)*rising_factorial(n,5)/factorial(6) for n in (1..40)] # G. C. Greubel, Aug 27 2019

Formula

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

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).

a(n) = binomial(n,6) + 2*binomial(n,5), n >= 5. - Zerinvary Lajos, Jul 26 2006

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

From G. C. Greubel, Aug 27 2019: (Start)

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

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

From Amiram Eldar, Feb 17 2023: (Start)

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

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

Extensions

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