A005715 Coefficient of x^7 in expansion of (1+x+x^2)^n.

4, 30, 126, 393, 1016, 2304, 4740, 9042, 16236, 27742, 45474, 71955, 110448, 165104, 241128, 344964, 484500, 669294, 910822, 1222749, 1621224, 2125200, 2756780, 3541590, 4509180, 5693454, 7133130, 8872231, 10960608, 13454496

Offset: 4

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 = 4..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 (8,-28,56,-70,56,-28,8,-1).

Crossrefs

Cf. A000574, A005581, A005712, A005714, A005716, A111808.

Programs

Magma

I:=[4, 30, 126, 393, 1016, 2304, 4740, 9042]; [n le 8 select I[n] else 8*Self(n-1)-28*Self(n-2)+56*Self(n-3)-70*Self(n-4)+56*Self(n-5)-28*Self(n-6)+8*Self(n-7)-Self(n-8): n in [1..40]]; // Vincenzo Librandi, Jun 16 2012

Magma

/* By definition: */ P:=PolynomialRing(Integers()); [ Coefficients((1+x+x^2)^n)[8]: n in [4..33] ]; // Bruno Berselli, Jun 17 2012

Maple

A005715:=(z-2)*(z**2-2)/(z-1)**8; # Conjectured by Simon Plouffe in his 1992 dissertation.
A005715 := n -> GegenbauerC(`if`(7A005715(n)), n=4..20); # Peter Luschny, May 10 2016

Mathematica

CoefficientList[Series[(x-2)*(x^2-2)/(1-x)^8,{x,0,40}],x] (* Vincenzo Librandi, Jun 16 2012 *)

Formula

a(n) = binomial(n, 4)*(n^3+27*n^2+116*n-120)/210, n >= 4.

G.f.: (x^4)*(x-2)*(x^2-2)/(1-x)^8. (Numerator polynomial is N3(7, x) from A063420).

a(n) = A027907(n, 7), n >= 4 (eighth column of trinomial coefficients).

a(n) = A111808(n,7) for n>6. - Reinhard Zumkeller, Aug 17 2005

a(n) = 8*a(n-1) -28*a(n-2) +56*a(n-3) -70*a(n-4) +56*a(n-5) -28*a(n-6) +8*a(n-7) -a(n-8). Vincenzo Librandi, Jun 16 2012

a(n) = 4*binomial(n,4) + 10*binomial(n,5) + 6*binomial(n,6) + binomial(n,7) (see our comment in A026729). - Vladimir Shevelev and Peter J. C. Moses, Jun 22 2012

a(n) = GegenbauerC(N, -n, -1/2) where N = 7 if 7Peter Luschny, May 10 2016

Extensions

More terms from Vladeta Jovovic, Oct 02 2000