A001815 a(n) = binomial(n,2) * 2^(n-1).

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

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• easy

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

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

Examples

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

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 and T. Mansour, Partially ordered patterns and their combinatorial interpretations, PU. M. A., Vol. 19, No. 2-3 (2008), pp. 27-38.
• INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 103.
• Milan Janjic, Enumerative Formulas for Some Functions on Finite Sets.
• Ran Pan and 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, arXiv:0911.4975 [math.NT], 2009.
• Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992.
• Aaron Robertson, Permutations containing and avoiding 123 and 132 patterns, Discrete Math. and Theoret. Computer Sci., Vol. 3, No. 4 (1999), pp. 151-154; arXiv preprint, arXiv:math/9903169 [math.CO], 1999.
• Aaron Robertson, Permutations restricted by two distinct patterns of length three, arXiv:math/0012029 [math.CO], 2000.
• Index entries for linear recurrences with constant coefficients, signature (6,-12,8).

Crossrefs

Cf. A001788, A089264, A090802.

Programs

Magma

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

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 *)
LinearRecurrence[{6,-12,8},{0,0,2},30] (* Harvey P. Dale, May 19 2018 *)

PARI

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

PARI

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

Sage

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

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(2*x). - 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 * binomial(n-1,i). - Wesley Ivan Hurt, Sep 20 2017

From Amiram Eldar, Jan 07 2022: (Start)

Sum_{n>=0} 1/a(n) = 2*(1 - log(2)).

Sum_{n>=0} (-1)^n/a(n) = 6*log(3/2) - 2. (End)