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A005715
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Coefficient of x^7 in expansion of (1+x+x^2)^n.
(Formerly M3632)
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6
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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
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OFFSET
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4,1
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REFERENCES
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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).
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LINKS
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FORMULA
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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) = 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) = GegenbauerC(N, -n, -1/2) where N = 7 if 7<n else 2*n-7. - Peter Luschny, May 10 2016
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MAPLE
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A005715 := n -> GegenbauerC(`if`(7<n, 7, 2*n-7), -n, -1/2):
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MATHEMATICA
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CoefficientList[Series[(x-2)*(x^2-2)/(1-x)^8, {x, 0, 40}], x] (* Vincenzo Librandi, Jun 16 2012 *)
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PROG
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(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<x>:=PolynomialRing(Integers()); [ Coefficients((1+x+x^2)^n)[8]: n in [4..33] ]; // Bruno Berselli, Jun 17 2012
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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