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A005716 Coefficient of x^8 in expansion of (1+x+x^2)^n.

%I A005716 M4975 #61 Feb 16 2025 08:32:29
%S A005716 1,15,90,357,1107,2907,6765,14355,28314,52624,93093,157950,258570,
%T A005716 410346,633726,955434,1409895,2040885,2903428,4065963,5612805,7646925,
%U A005716 10293075,13701285,18050760,23554206,30462615,39070540,49721892,62816292,78816012,98253540
%N A005716 Coefficient of x^8 in expansion of (1+x+x^2)^n.
%D A005716 L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 78.
%D A005716 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H A005716 Vincenzo Librandi, <a href="/A005716/b005716.txt">Table of n, a(n) for n = 4..1000</a>
%H A005716 R. K. Guy, <a href="/A005712/a005712.pdf">Letter to N. J. A. Sloane, 1987</a>
%H A005716 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 A005716 Simon Plouffe, <a href="/A000051/a000051_2.pdf">1031 Generating Functions</a>, Appendix to Thesis, Montreal, 1992
%H A005716 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/TrinomialCoefficient.html">Trinomial Coefficient</a>
%H A005716 <a href="/index/Rec#order_09">Index entries for linear recurrences with constant coefficients</a>, signature (9,-36,84,-126,126,-84,36,-9,1).
%F A005716 a(n) = binomial(n+1, 5)*(n^2+23*n-84)*(n+10)/336, n >= 4.
%F A005716 G.f.: (x^4)*(1+6*x-9*x^2+3*x^3)/(1-x)^9. (Numerator polynomial is N3(8, x) from A063420).
%F A005716 a(n) = A027907(n, 8), n >= 4 (ninth column of trinomial coefficients).
%F A005716 a(n) = A111808(n,8) for n>7. - _Reinhard Zumkeller_, Aug 17 2005
%F A005716 a(n) = 9*a(n-1) - 36*a(n-2) + 84*a(n-3) - 126*a(n-4) + 126*a(n-5) - 84*a(n-6) + 36*a(n-7) - 9*a(n-8) + a(n-9). _Vincenzo Librandi_, Jun 16 2012
%F A005716 a(n) = binomial(n,4) + 10*binomial(n,5) + 15*binomial(n,6) + 7*binomial(n,7) + binomial(n,8) (see our comment in A026729). - _Vladimir Shevelev_ and _Peter J. C. Moses_, Jun 22 2012
%F A005716 a(n) = GegenbauerC(N, -n, -1/2) where N = 8 if 8<n else 2*n-8. - _Peter Luschny_, May 10 2016
%p A005716 A005716:=-(6*z-9*z**2+3*z**3+1)/(z-1)**9; # Conjectured by _Simon Plouffe_ in his 1992 dissertation.
%p A005716 A005716 := n -> GegenbauerC(`if`(8<n,8,2*n-8), -n, -1/2):
%p A005716 seq(simplify(A005716(n)), n=4..20); # _Peter Luschny_, May 10 2016
%t A005716 CoefficientList[Series[(1+6*x-9*x^2+3*x^3)/(1-x)^9,{x,0,40}],x] (* _Vincenzo Librandi_, Jun 16 2012 *)
%o A005716 (Magma) I:=[1, 15, 90, 357, 1107, 2907, 6765, 14355, 28314]; [n le 9 select I[n] else 9*Self(n-1)-36*Self(n-2)+84*Self(n-3)-126*Self(n-4)+126*Self(n-5)-84*Self(n-6)+36*Self(n-7)-9*Self(n-8)+Self(n-9): n in [1..40]]; // _Vincenzo Librandi_, Jun 16 2012
%o A005716 (Magma) /* By definition: */ P<x>:=PolynomialRing(Integers()); [ Coefficients((1+x+x^2)^n)[9]: n in [4..32] ]; // _Bruno Berselli_, Jun 17 2012
%Y A005716 Cf. A000574, A005581, A005712, A005714, A005715, A111808.
%K A005716 nonn,easy
%O A005716 4,2
%A A005716 _N. J. A. Sloane_
%E A005716 More terms from _Vladeta Jovovic_, Oct 02 2000