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
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
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).
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)
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
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 *)
PROG
(Sage) [lucas_number1(n, 2, 0)*binomial(n, 2) for n in range(0, 29)] # Zerinvary Lajos, Mar 10 2009
(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
(Magma) [Binomial(n, 2)*2^(n-1): n in [0..30]]; // Vincenzo Librandi, Mar 14 2014
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
STATUS
approved