A005714 Coefficient of x^6 in expansion of (1+x+x^2)^n.

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, 4193322, 4908309, 5721717

Offset: 3

Author

N. J. A. Sloane

Keywords

• nonn
• easy

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; arXiv:0911.4975 [math.NT], 2009.
• Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 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).

Crossrefs

Cf. A000574, A005581, A005712, A005715-A005716, A111808.

Programs

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

Magma

/* By definition: */ P:=PolynomialRing(Integers()); [ Coefficients((1+x+x^2)^n)[7]: n in [3..35] ]; // Bruno Berselli, Jun 17 2012

Maple

A005714:=-(1+3*z-4*z**2+z**3)/(z-1)**7; # Conjectured by Simon Plouffe in his 1992 dissertation.
A005714 := n -> GegenbauerC(`if`(6A005714(n)), n=3..20); # Peter Luschny, May 10 2016

Mathematica

a[n_] := Coefficient[(1 + x + x^2)^n, x, 6]; Table[a[n], {n, 3, 35}]
CoefficientList[Series[(1+3*x-4*x^2+x^3)/(1-x)^7,{x,0,40}],x] (* Vincenzo Librandi, Jun 16 2012 *)

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 6Peter Luschny, May 10 2016

E.g.f.: exp(x)*x^3*(120 + 180*x + 30*x^2 + x^3)/720. - Stefano Spezia, Mar 28 2023

Extensions

More terms from Vladeta Jovovic, Oct 02 2000