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A005286
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(n + 3)*(n^2 + 6*n + 2)/6.
(Formerly M4109)
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4
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1, 6, 15, 29, 49, 76, 111, 155, 209, 274, 351, 441, 545, 664, 799, 951, 1121, 1310, 1519, 1749, 2001, 2276, 2575, 2899, 3249, 3626, 4031, 4465, 4929, 5424, 5951, 6511, 7105, 7734, 8399, 9101, 9841, 10620, 11439, 12299, 13201, 14146, 15135, 16169, 17249
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,2
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COMMENTS
| Number of permutations of [n+3] with three inversions. - Michael Somos, Jun 25, 2002
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REFERENCES
| L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 255, #2, b(n,3).
R. K. Guy, personal communication.
R. H. Moritz and R. C. Williams, A coin-tossing problem and some related combinatorics, Math. Mag., 61 (1988), 24-29.
E. Netto, Lehrbuch der Combinatorik. 2nd ed., Teubner, Leipzig, 1927, p. 96.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
R. P. Stanley, Enumerative Combinatorics, Cambridge, Vol. 1, 1999; see Exercise 1.30, p. 49.
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LINKS
| T. D. Noe, Table of n, a(n) for n=0..1000
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.
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FORMULA
| a(-6-n)=-a(n). - Michael Somos May 12 2005
binomial(n,3)+binomial(n,2)-n, n>=3. - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Jul 23 2006
G.f.: (1+2*x-3*x^2+x^3)/(1-x)^4. a(n) = a(n-1)+A000096(n+1) = A005581(n+2)-1. - Henry Bottomley, Oct 25 2001
(m^3-7*m)/6 for m >= 3 gives the same sequence. - N. J. A. Sloane, Jul 15 2011
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MAPLE
| [seq(binomial(n, 3)+binomial(n, 2)-n, n=3..45)]; - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Jul 23 2006
A005286:=(1+2*z-3*z**2+z**3)/(z-1)**4; [Conjectured by S. Plouffe in his 1992 dissertation.]
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MATHEMATICA
| Table[(n + 3) (n^2 + 6*n + 2)/6, {n, 0, 100}] (* From Vladimir Joseph Stephan Orlovsky, Jul 16 2011 *)
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PROG
| (PARI) a(n)=n+=3; (n^3-7*n)/6 /* Michael Somos May 12 2005 */
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CROSSREFS
| Cf. A008302, A193106, A193107.
Sequence in context: A066995 A180953 A200184 * A025212 A024972 A048749
Adjacent sequences: A005283 A005284 A005285 * A005287 A005288 A005289
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KEYWORD
| nonn,easy,nice
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com).
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