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A005584 Coefficients of Chebyshev polynomials.
(Formerly M2059)
7
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 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,1
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
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].
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. [Cached copy, May 15 2013]
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)
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 *)
PROG
(PARI) a(n)=n*(n+11)*(n+4)*(n+3)*(n+2)*(n+1)/720 \\ Charles R Greathouse IV, Jun 14 2011
(Magma) [n*(n+11)*(n+4)*(n+3)*(n+2)*(n+1)/720: n in [1..40]]; // Vincenzo Librandi, Jun 15 2011
(Sage) [(n+11)*rising_factorial(n, 5)/factorial(6) for n in (1..40)] # G. C. Greubel, Aug 27 2019
(GAP) List([1..40], n-> (n+11)*Binomial(n+4, 5)/6); # G. C. Greubel, Aug 27 2019
CROSSREFS
Sequence in context: A333519 A270294 A330539 * A072416 A319759 A254784
KEYWORD
nonn,easy
AUTHOR
EXTENSIONS
More terms from Klaus Strassburger (strass(AT)ddfi.uni-duesseldorf.de), Dec 07 1999
STATUS
approved

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Last modified March 19 02:49 EDT 2024. Contains 370952 sequences. (Running on oeis4.)