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A005716
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Coefficient of x^8 in expansion of (1+x+x^2)^n
(Formerly M4975)
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12
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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
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OFFSET
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4,2
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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+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) = 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) = GegenbauerC(N, -n, -1/2) where N = 8 if 8<n else 2*n-8. - Peter Luschny, May 10 2016
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MAPLE
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A005716 := n -> GegenbauerC(`if`(8<n, 8, 2*n-8), -n, -1/2):
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MATHEMATICA
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CoefficientList[Series[(1+6*x-9*x^2+3*x^3)/(1-x)^9, {x, 0, 40}], x] (* Vincenzo Librandi, Jun 16 2012 *)
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PROG
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(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
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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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