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A000574
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Coefficient of x^5 in expansion of (1 + x + x^2)^n.
(Formerly M3011 N1219)
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12
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3, 16, 51, 126, 266, 504, 882, 1452, 2277, 3432, 5005, 7098, 9828, 13328, 17748, 23256, 30039, 38304, 48279, 60214, 74382, 91080, 110630, 133380, 159705, 190008, 224721, 264306, 309256, 360096, 417384, 481712, 553707, 634032, 723387, 822510
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OFFSET
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3,1
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COMMENTS
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If Y is a 3-subset of an n-set X then, for n>=7, a(n-4) is the number of 5-subsets of X having at most one element in common with Y. - Milan Janjic, Nov 23 2007
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REFERENCES
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L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 78.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
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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G.f.: x^3*(3-2*x)/(1-x)^6.
a(n) = 3*binomial(n+2,5) - 2*binomial(n+1,5).
a(n) = binomial(n+1, 4)*(n+12)/5 = 3*b(n-3)-2*b(n-4), with b(n)=binomial(n+5, 5); cf. A000389.
a(n) = 6*a(n-1) - 15*a(n-2) + 20*a(n-3) - 15*a(n-4) + 6*a(n-5) - a(n-6). - Vincenzo Librandi, Jun 10 2012
a(n) = GegenbauerC(N, -n, -1/2) where N = 5 if 5<n else 2*n-5. - Peter Luschny, May 10 2016
E.g.f.: exp(x)*x^3*(60 + 20*x + x^2)/120. - Stefano Spezia, Jul 09 2023
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MAPLE
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seq(3*binomial(n+2, 5)-2*binomial(n+1, 5), n=3..100); # Robert Israel, Aug 04 2015
A000574 := n -> GegenbauerC(`if`(5<n, 5, 2*n-5), -n, -1/2):
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MATHEMATICA
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CoefficientList[Series[(3-2*x)/(1-x)^6, {x, 0, 40}], x] (* Vincenzo Librandi, Jun 10 2012 *)
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PROG
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(Magma) [3*Binomial(n+2, 5)-2*Binomial(n+1, 5): n in [3..50]]; // Vincenzo Librandi, Jun 10 2012
(PARI) x='x+O('x^50); Vec(x^3*(3-2*x)/(1-x)^6) \\ G. C. Greubel, Nov 22 2017
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CROSSREFS
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Column m=5 of (1, 3) Pascal triangle A095660.
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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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