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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 (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 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]. R. 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. Simon Plouffe, 1031 Generating Functions and Conjectures, Université du Québec à Montréal, 1992. Index entries for linear recurrences with constant coefficients, signature (7,-21,35,-35,21,-7,1). 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) 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[Binomial[n, 5] + Binomial[n, 6], {n, 40}] (* Vladimir Joseph Stephan Orlovsky, Jun 14 2011, modified by G. C. Greubel, Aug 27 2019 *) Table[(n+11)*Pochhammer[5, n]/30, {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 Cf. A127672, A005581, A005582, A005583. Sequence in context: A176060 A168172 A270294 * A072416 A319759 A254784 Adjacent sequences:  A005581 A005582 A005583 * A005585 A005586 A005587 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 October 17 21:16 EDT 2019. Contains 328132 sequences. (Running on oeis4.)