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A001815
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C(n,2)*2^(n-1).
(Formerly M2021 N0799)
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10
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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; refs; listen; history; internal format)
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OFFSET
| 0,3
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COMMENTS
| 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]
Consider all binary words of length n, and assign a weight to each set bit - the leftmost gets a weight of n-1, the rightmost a weight of 0. a(n) gives the sum
of the weights of all n-bit words. For example, if n=3, we have 000,001,010,011,100,101,110,111 with weights of 0,0,1,1,2,2,3,3, giving a sum of 12.
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REFERENCES
| 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.
A. Robertson, Permutations containing and avoiding 123 and 132 patterns, Discrete Math. and Theoret. Computer Sci., 3 (1999), 151-154.
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
| 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].
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 103
Milan Janjic, Enumerative Formulas for Some Functions on Finite Sets
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.
A. Robertson, Permutations restricted by two distinct patterns of length three
A. Robertson, Permutations containing and avoiding 123 and 132 patterns
Index to sequences with linear recurrences with constant coefficients, signature (6,-12,8).
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FORMULA
| 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]
a(n) = 2*a(n-1) + n*2^n
G.f.: 2*x^2/(1 - 2*x)^3. [Arkadiusz Wesolowski, Dec 21 2011]
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MAPLE
| 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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MATHEMATICA
| Table[Binomial[n, 2]*2^(n - 1), {n, 0, 28}] (* Arkadiusz Wesolowski, Dec 21 2011 *)
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PROG
| (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]
(PARI) a(n)=binomial(n, 2)<<(n-1) \\ Charles R Greathouse IV, Dec 21 2011
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CROSSREFS
| Cf. A089264.
Equals A090802(n, 2).
Sequence in context: A069946 A176684 A048501 * A052569 A052591 A029766
Adjacent sequences: A001812 A001813 A001814 * A001816 A001817 A001818
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KEYWORD
| nonn,easy
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com).
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