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A005714
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Coefficient of x^6 in expansion of (1+x+x^2)^n.
(Formerly M4704)
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9
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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
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OFFSET
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3,2
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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, 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) = 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) = GegenbauerC(N, -n, -1/2) where N = 6 if 6<n else 2*n-6. - Peter 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
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MAPLE
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A005714 := n -> GegenbauerC(`if`(6<n, 6, 2*n-6), -n, -1/2):
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MATHEMATICA
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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 *)
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PROG
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(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<x>:=PolynomialRing(Integers()); [ Coefficients((1+x+x^2)^n)[7]: n in [3..35] ]; // 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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