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A005716 Coefficient of x^8 in expansion of (1+x+x^2)^n
(Formerly M4975)
12
1, 15, 90, 357, 1107, 2907, 6765, 14355, 28314, 52624, 93093, 157950, 258570, 410346, 633726, 955434, 1409895, 2040885, 2903428, 4065963, 5612805, 7646925, 10293075, 13701285, 18050760, 23554206, 30462615, 39070540, 49721892 (list; graph; refs; listen; history; text; internal format)
OFFSET

4,2

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 = 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.

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 (9,-36,84,-126,126,-84,36,-9,1).

FORMULA

a(n) = binomial(n+1, 5)*(n^2+23*n-84)*(n+10)/336, n >= 4.

G.f.: (x^4)*(1+6*x-9*x^2+3*x^3)/(1-x)^9 (Numerator polynomial is N3(8, x) from A063420).

a(n) = A027907(n, 8), n >= 4 (ninth column of trinomial coefficients).

a(n) = A111808(n,8) for n>7. - Reinhard Zumkeller, Aug 17 2005

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

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

a(n) = GegenbauerC(N, -n, -1/2) where N = 8 if 8<n else 2*n-8. - Peter Luschny, May 10 2016

MAPLE

A005716:=-(6*z-9*z**2+3*z**3+1)/(z-1)**9; # Conjectured by Simon Plouffe in his 1992 dissertation.

A005716 := n -> GegenbauerC(`if`(8<n, 8, 2*n-8), -n, -1/2):

seq(simplify(A005716(n)), n=4..20); # Peter Luschny, May 10 2016

MATHEMATICA

CoefficientList[Series[(1+6*x-9*x^2+3*x^3)/(1-x)^9, {x, 0, 40}], x] (* Vincenzo Librandi, Jun 16 2012 *)

PROG

(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

(MAGMA) /* By definition: */ P<x>:=PolynomialRing(Integers()); [ Coefficients((1+x+x^2)^n)[9]: n in [4..32] ]; // Bruno Berselli, Jun 17 2012

CROSSREFS

Cf. A000574, A005581, A005712, A005714, A005715, A111808.

Sequence in context: A151974 A179096 A001297 * A292055 A263628 A263629

Adjacent sequences:  A005713 A005714 A005715 * A005717 A005718 A005719

KEYWORD

nonn,easy

AUTHOR

N. J. A. Sloane.

EXTENSIONS

More terms from Vladeta Jovovic, Oct 02 2000

STATUS

approved

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Last modified February 23 09:47 EST 2018. Contains 299520 sequences. (Running on oeis4.)