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Search: id:A000574
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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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+0
9
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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
(list; graph; listen)
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OFFSET
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3,1
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COMMENT
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G.f.: x^3*(3-2*x)/(1-x)^6. 3*binomial(n+2,5)-2*binomial(n+1,5)
a(n) = A111808(n,5) for n>4. - Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), Aug 17 2005
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 R. Janjic (agnus(AT)blic.net), Nov 23 2007
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REFERENCES
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N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
S. Plouffe, Approximations de S\'{e}ries G\'{e}n\'{e}ratrices et Quelques Conjectures, Dissertation, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al, 1992.
L. Carlitz et al., Permutations and sequences with repetions by number of increases, J. Combin. Theory, 1 (1966), 350-374.
L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 78.
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LINKS
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S. Plouffe, Approximations de S\'{e}ries G\'{e}n\'{e}ratrices et Quelques Conjectures, Dissertation, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al, 1992.
S. Plouffe, 1031 Generating Functions and Conjectures, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al, 1992.
Eric Weisstein's World of Mathematics, Trinomial Coefficient
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FORMULA
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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.
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MAPLE
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A000574:=-(-3+2*z)/(z-1)**6; [Conjectured by S. Plouffe in his 1992 dissertation.]
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CROSSREFS
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Cf. A005581, A005712, A005714-A005716.
Column m=5 of (1, 3) Pascal triangle A095660.
Cf. A005712, A000581.
Sequence in context: A152618 A004320 A089363 this_sequence A041233 A055194 A027540
Adjacent sequences: A000571 A000572 A000573 this_sequence A000575 A000576 A000577
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KEYWORD
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nonn,easy
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AUTHOR
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N. J. A. Sloane (njas(AT)research.att.com).
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EXTENSIONS
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More terms from Vladeta Jovovic (vladeta(AT)eunet.rs), Oct 02 2000
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