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%I M4704 #63 Jun 28 2023 20:39:13
%S 1,10,45,141,357,784,1554,2850,4917,8074,12727,19383,28665,41328,
%T 58276,80580,109497,146490,193249,251713,324093,412896,520950,651430,
%U 807885,994266,1214955,1474795,1779121,2133792,2545224,3020424,3567025,4193322,4908309,5721717
%N Coefficient of x^6 in expansion of (1+x+x^2)^n.
%D L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 78.
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H Vincenzo Librandi, <a href="/A005714/b005714.txt">Table of n, a(n) for n = 3..1000</a>
%H R. K. Guy, <a href="/A005712/a005712.pdf">Letter to N. J. A. Sloane, 1987</a>
%H Simon Plouffe, <a href="https://arxiv.org/abs/0911.4975">Approximations de séries génératrices et quelques conjectures</a>, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
%H Simon Plouffe, <a href="/A000051/a000051_2.pdf">1031 Generating Functions</a>, Appendix to Thesis, Montreal, 1992
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/TrinomialCoefficient.html">Trinomial Coefficient</a>
%H <a href="/index/Rec#order_07">Index entries for linear recurrences with constant coefficients</a>, signature (7,-21,35,-35,21,-7,1).
%F a(n) = binomial(n, 3)*(n^3+18*n^2+17*n-120) /120.
%F G.f.: (x^3)*(1+3*x-4*x^2+x^3)/(1-x)^7. (Numerator polynomial is N3(6, x) from A063420).
%F a(n) = A027907(n, 6), n >= 3 (seventh column of trinomial coefficients).
%F a(n) = A111808(n,6) for n>5. - _Reinhard Zumkeller_, Aug 17 2005
%F 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
%F 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
%F a(n) = GegenbauerC(N, -n, -1/2) where N = 6 if 6<n else 2*n-6. - _Peter Luschny_, May 10 2016
%F E.g.f.: exp(x)*x^3*(120 + 180*x + 30*x^2 + x^3)/720. - _Stefano Spezia_, Mar 28 2023
%p A005714:=-(1+3*z-4*z**2+z**3)/(z-1)**7; # Conjectured by _Simon Plouffe_ in his 1992 dissertation.
%p A005714 := n -> GegenbauerC(`if`(6<n,6,2*n-6), -n, -1/2):
%p seq(simplify(A005714(n)), n=3..20); # _Peter Luschny_, May 10 2016
%t a[n_] := Coefficient[(1 + x + x^2)^n, x, 6]; Table[a[n], {n, 3, 35}]
%t CoefficientList[Series[(1+3*x-4*x^2+x^3)/(1-x)^7,{x,0,40}],x] (* _Vincenzo Librandi_, Jun 16 2012 *)
%o (Magma) I:=[1, 10, 45, 141, 357, 784, 1554]; [n le 7 select I[n] else 7*Self(n-1)-21*Self(n-2)+35*Self(n-3)-35*Self(n-4)+21*Self(n-5)-7*Self(n-6)+Self(n-7): n in [1..40]]; // _Vincenzo Librandi_, Jun 16 2012 *)
%o (Magma) /* By definition: */ P<x>:=PolynomialRing(Integers()); [ Coefficients((1+x+x^2)^n)[7]: n in [3..35] ]; // _Bruno Berselli_, Jun 17 2012
%Y Cf. A000574, A005581, A005712, A005715-A005716, A111808.
%K nonn,easy
%O 3,2
%A _N. J. A. Sloane_
%E More terms from _Vladeta Jovovic_, Oct 02 2000
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