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 A005714 Coefficient of x^6 in expansion of (1+x+x^2)^n. (Formerly M4704) 9
 1, 10, 45, 141, 357, 784, 1554, 2850, 4917, 8074, 12727, 19383, 28665, 41328, 58276, 80580, 109497, 146490, 193249, 251713, 324093, 412896, 520950, 651430, 807885, 994266, 1214955, 1474795, 1779121, 2133792, 2545224, 3020424, 3567025 (list; graph; refs; listen; history; text; internal format)
 OFFSET 3,2 REFERENCES L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 78. 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 = 3..1000 R. K. Guy, Letter to N. J. A. Sloane, 1987 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. Eric Weisstein's World of Mathematics, Trinomial Coefficient Index entries for linear recurrences with constant coefficients, signature (7,-21,35,-35,21,-7, 1). FORMULA a(n) = binomial(n, 3)*(n^3+18*n^2+17*n-120) /120. G.f.: (x^3)*(1+3*x-4*x^2+x^3)/(1-x)^7 (Numerator polynomial is N3(6, x) from A063420). a(n) = A027907(n, 6), n >= 3 (seventh column of trinomial coefficients). a(n) = A111808(n,6) for n>5. - Reinhard Zumkeller, Aug 17 2005 a(n) = 7*a(n-1) -21*a(n-2) +35*a(n-3) -35*a(n-4) +21*a(n-5) -7*a(n-6) +a(n-7). Vincenzo Librandi, Jun 16 2012 a(n) = binomial(n,3) + 6*binomial(n,4) + 5*binomial(n,5) + binomial(n,6) (see our comment in A026729). - Vladimir Shevelev and Peter J. C. Moses, Jun 22 2012 a(n) = GegenbauerC(N, -n, -1/2) where N = 6 if 6 GegenbauerC(`if`(6:=PolynomialRing(Integers()); [ Coefficients((1+x+x^2)^n)[7]: n in [3..35] ]; // Bruno Berselli, Jun 17 2012 CROSSREFS Cf. A000574, A005581, A005712, A005715-A005716, A111808. Sequence in context: A213188 A037270 A027800 * A175705 A143671 A221532 Adjacent sequences:  A005711 A005712 A005713 * A005715 A005716 A005717 KEYWORD nonn,easy AUTHOR EXTENSIONS More terms from Vladeta Jovovic, Oct 02 2000 STATUS approved

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Last modified October 16 00:50 EDT 2018. Contains 316252 sequences. (Running on oeis4.)