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A001815 a(n) = C(n,2) * 2^(n-1).
(Formerly M2021 N0799)
16
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, 108984795136 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

Number of permutations of length n+3 containing 132 and 123 exactly once. Likewise for the 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. [Geoffrey Critzer, 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.

a(n) is the number of North-East paths from (0,0) to (n+2,n+2) that have exactly one east step below y = x-1 and exactly one east step above y = x+1. This is related to the paired pattern P_1 and P_2. More details can be found in Pan and Remmel's link. - Ran Pan, Feb 03 2016

a(n) is the number of diagonals of length sqrt(2) in an n-dimensional hypercube (same as diagonals of its two-dimensional faces). - Stanislav Sykora, Oct 23 2016

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.

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).

LINKS

T. D. Noe, Table of n, a(n) for n = 0..500

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].

A. Burstein, S. Kitaev, T. Mansour, Partially ordered patterns and their combinatorial interpretations, PU. M. A. Vol. 19 (2008), No. 2-3, pp. 27-38.

INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 103

Milan Janjic, Enumerative Formulas for Some Functions on Finite Sets

Ran Pan, Jeffrey B. Remmel, Paired patterns in lattice paths, arXiv:1601.07988 [math.CO], 2016.

Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992.

Simon Plouffe, 1031 Generating Functions and Conjectures, Université du Québec à Montréal, 1992.

A. Robertson, Permutations restricted by two distinct patterns of length three, arXiv:math/0012029 [math.CO], 2000.

A. Robertson, Permutations containing and avoiding 123 and 132 patterns

A. Robertson, Permutations containing and avoiding 123 and 132 patterns, Discrete Math. and Theoret. Computer Sci., 3 (1999), 151-154.

A. Robertson, Permutations containing and avoiding 123 and 132 patterns, arXiv:math/9903169 [math.CO], 1999.

Index entries for linear recurrences with constant coefficients, signature (6,-12,8).

FORMULA

G.f.: 2*x^2/(1 - 2*x)^3. [Simon Plouffe in his 1992 dissertation]

a(n) = A090802(n, 2).

a(n) = Sum_{i=0..n} i*(n-i)*binomial(n, i). - Benoit Cloitre, Nov 11 2004

a(n) = Sum_{k=0..n} k*2^(k-1). - Zerinvary Lajos, Oct 09 2006

a(n) = Sum_{j=0..n} binomial(n-1,j)*n*j. - Zerinvary Lajos, Oct 19 2006

E.g.f.: x^2*exp(2x). [Geoffrey Critzer, Apr 07 2009]

a(n) = 2^(n-2)*n*(n-1). [Tobias Friedrich (tfried(AT)mpi-inf.mpg.de), Jun 18 2009]

a(n) = 2*a(n-1) + n*2^n.

For n>0, a(n) = 2*A001788(n-1). - Stanislav Sykora, Oct 23 2016

a(n) = a(1-n) * 2^(2*n-1) for all n in Z. - Michael Somos, Oct 25 2016

a(n) = Sum_{k=0..n-1} Sum_{i=0..n-1} k * C(n-1,i). - Wesley Ivan Hurt, Sep 20 2017

EXAMPLE

G.f. = 2*x^2 + 12*x^3 + 48*x^4 + 160*x^5 + 480*x^6 + 1344*x^7 + 3584*x^8 + ...

MAPLE

A001815 := proc(n)

    2^(n-2)*n*(n-1) ;

end proc: # R. J. Mathar, Mar 12 2014

MATHEMATICA

Table[Binomial[n, 2]*2^(n-1), {n, 0, 28}] (* Arkadiusz Wesolowski, Dec 21 2011 *)

CoefficientList[Series[2 x^2/(1 - 2 x)^3, {x, 0, 40}], x] (* Vincenzo Librandi, Mar 14 2014 *)

PROG

(Sage) [lucas_number1(n, 2, 0)*binomial(n, 2) for n in xrange(0, 29)] # Zerinvary Lajos, Mar 10 2009

(PARI) a(n)=binomial(n, 2)<<(n-1) \\ Charles R Greathouse IV, Dec 21 2011

(MAGMA) [Binomial(n, 2)*2^(n-1): n in [0..30]]; // Vincenzo Librandi, Mar 14 2014

(PARI) x='x+O('x^100); concat([0, 0], Vec(2*x^2/(1-2*x)^3)) \\ Altug Alkan, Nov 01 2015

CROSSREFS

Cf. A001788, A089264, A090802.

Sequence in context: A176684 A216152 A048501 * A052569 A221663 A232663

Adjacent sequences:  A001812 A001813 A001814 * A001816 A001817 A001818

KEYWORD

nonn,easy,changed

AUTHOR

N. J. A. Sloane

STATUS

approved

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Last modified September 26 06:54 EDT 2017. Contains 292502 sequences.