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A001815
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a(n) = binomial(n,2) * 2^(n-1).
(Formerly M2021 N0799)
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21
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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, 108984795136
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OFFSET
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0,3
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COMMENTS
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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
a(n) is the number of ways to select a team from n players with at least two players, two of whom are the captain and the goalkeeper. - Wojciech Raszka, Apr 10 2019
a(n) is the sum of N_0*N_1 for all binary strings of length n, where N_0 and N_1 are the number of 0's and 1's in the string, respectively. For example, if n=3, we have 000, 001, 010, 011, 100, 101, 110, 111 with products 0, 2, 2, 2, 2, 2, 2, 0, giving a sum of 12. - Sigurd Kittilsen and Jens Otten, Sep 17 2020
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REFERENCES
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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).
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LINKS
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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].
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FORMULA
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G.f.: 2*x^2/(1 - 2*x)^3. [Simon Plouffe in his 1992 dissertation]
a(n) = Sum_{i=0..n} i*(n-i)*binomial(n, i). - Benoit Cloitre, Nov 11 2004
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.
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 * binomial(n-1,i). - Wesley Ivan Hurt, Sep 20 2017
Sum_{n>=0} 1/a(n) = 2*(1 - log(2)).
Sum_{n>=0} (-1)^n/a(n) = 6*log(3/2) - 2. (End)
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EXAMPLE
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G.f. = 2*x^2 + 12*x^3 + 48*x^4 + 160*x^5 + 480*x^6 + 1344*x^7 + 3584*x^8 + ...
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MAPLE
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2^(n-2)*n*(n-1) ;
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MATHEMATICA
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CoefficientList[Series[2 x^2/(1 - 2 x)^3, {x, 0, 40}], x] (* Vincenzo Librandi, Mar 14 2014 *)
LinearRecurrence[{6, -12, 8}, {0, 0, 2}, 30] (* Harvey P. Dale, May 19 2018 *)
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PROG
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(Sage) [lucas_number1(n, 2, 0)*binomial(n, 2) for n in range(0, 29)] # Zerinvary Lajos, Mar 10 2009
(PARI) my(x='x+O('x^100)); concat([0, 0], Vec(2*x^2/(1-2*x)^3)) \\ Altug Alkan, Nov 01 2015
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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STATUS
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approved
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