

A001815


a(n) = binomial(n,2) * 2^(n1).
(Formerly M2021 N0799)


19



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

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 n1, the rightmost a weight of 0. a(n) gives the sum of the weights of all nbit 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 NorthEast paths from (0,0) to (n+2,n+2) that have exactly one east step below y = x1 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 ndimensional hypercube (same as diagonals of its twodimensional 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


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

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].


FORMULA

G.f.: 2*x^2/(1  2*x)^3. [Simon Plouffe in his 1992 dissertation]
a(n) = Sum_{i=0..n} i*(ni)*binomial(n, i).  Benoit Cloitre, Nov 11 2004
a(n) = 2^(n2)*n*(n1).  Tobias Friedrich (tfried(AT)mpiinf.mpg.de), Jun 18 2009
a(n) = 2*a(n1) + n*2^n.
a(n) = a(1n) * 2^(2*n1) for all n in Z.  Michael Somos, Oct 25 2016
a(n) = Sum_{k=0..n1} Sum_{i=0..n1} k * binomial(n1,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)


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

2^(n2)*n*(n1) ;


MATHEMATICA

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 *)


PROG

(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/(12*x)^3)) \\ Altug Alkan, Nov 01 2015


CROSSREFS



KEYWORD

nonn,easy


AUTHOR



STATUS

approved



