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Search: id:A001815
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| A001815 |
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C(n,2)*2^(n-1).
(Formerly M2021 N0799)
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+0
8
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| 0, 0, 2, 12, 48, 160, 480, 1344, 3584, 9216, 23040, 56320, 135168, 319488, 745472, 1720320, 3932160, 8912896, 20054016, 44826624, 99614720, 220200960, 484442112, 1061158912, 2315255808, 5033164800, 10905190400, 23555211264, 50734301184
(list; graph; listen)
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OFFSET
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0,3
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COMMENT
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Number of permutations of length n+3 containing exactly once 132 and 123. Likewise for pattern pairs (123,213), (231,321), (312,321).
a(n) is the number of ways to assign n distinct contestants to two (not necessarily equal) distinct teams and then choose a captain for each team. [From Geoffrey Critzer (critzer.geoffrey(AT)usd443.org), Apr 07 2009]
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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.
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 801.
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LINKS
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Milan Janjic, Enumerative Formulas for Some Functions on Finite Sets
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
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.
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 103
A. Robertson, Permutations restricted by two distinct patterns of length three
A. Robertson, Permutations containing and avoiding 123 and 132 patterns
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FORMULA
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a(n) = sum(i=0, n, i*(n-i)*binomial(n, i)) - Benoit Cloitre (benoit7848c(AT)orange.fr), Nov 11 2004
a(n)=sum(k*2^(k-1), k=0..n). - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Oct 09 2006
Sum(binomial(n-1,j)*n*j,j=0..n). - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Oct 19 2006
E.g.f.:x^2*exp(2x) [From Geoffrey Critzer (critzer.geoffrey(AT)usd443.org), Apr 07 2009]
a(n) = 2^n*n*(n-1)/4 [From Tobias Friedrich (tfried(AT)mpi-inf.mpg.de), Jun 18 2009]
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MAPLE
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a:=n->sum(binomial(n-1, j)*n*j, j=0..n): seq(a(n), n=0..28); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Oct 19 2006
A001815:=-2*z/(2*z-1)**3; [Conjectured by S. Plouffe in his 1992 dissertation.]
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PROGRAM
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(Other) SAGE: [lucas_number1(n, 2, 0)*binomial(n, 2) for n in xrange(0, 29)] [From Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Mar 10 2009]
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CROSSREFS
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Cf. A089264.
Equals A090802(n, 2).
Sequence in context: A129018 A069946 A048501 this_sequence A052569 A052591 A029766
Adjacent sequences: A001812 A001813 A001814 this_sequence A001816 A001817 A001818
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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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