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%I A001815 M2021 N0799 #126 Sep 08 2022 08:44:29
%S A001815 0,0,2,12,48,160,480,1344,3584,9216,23040,56320,135168,319488,745472,
%T A001815 1720320,3932160,8912896,20054016,44826624,99614720,220200960,
%U A001815 484442112,1061158912,2315255808,5033164800,10905190400,23555211264,50734301184,108984795136
%N A001815 a(n) = binomial(n,2) * 2^(n-1).
%C A001815 Number of permutations of length n+3 containing 132 and 123 exactly once. Likewise for the pairs (123,213), (231,321), (312,321).
%C A001815 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
%C A001815 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.
%C A001815 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
%C A001815 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
%C A001815 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
%C A001815 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
%D A001815 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.
%D A001815 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%D A001815 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H A001815 T. D. Noe, <a href="/A001815/b001815.txt">Table of n, a(n) for n = 0..500</a>
%H A001815 M. Abramowitz and I. A. Stegun, eds., <a href="http://www.convertit.com/Go/ConvertIt/Reference/AMS55.ASP">Handbook of Mathematical Functions</a>, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
%H A001815 A. Burstein, S. Kitaev and T. Mansour, <a href="http://puma.dimai.unifi.it/19_2_3/3.pdf">Partially ordered patterns and their combinatorial interpretations</a>, PU. M. A., Vol. 19, No. 2-3 (2008), pp. 27-38.
%H A001815 INRIA Algorithms Project, <a href="http://ecs.inria.fr/services/structure?nbr=103">Encyclopedia of Combinatorial Structures 103</a>.
%H A001815 Milan Janjic, <a href="https://pmf.unibl.org/wp-content/uploads/2017/10/enumfun.pdf">Enumerative Formulas for Some Functions on Finite Sets</a>.
%H A001815 Ran Pan and Jeffrey B. Remmel, <a href="http://arxiv.org/abs/1601.07988">Paired patterns in lattice paths</a>, arXiv:1601.07988 [math.CO], 2016.
%H A001815 Simon Plouffe, <a href="https://arxiv.org/abs/0911.4975">Approximations de séries génératrices et quelques conjectures</a>, Dissertation, Université du Québec à Montréal, 1992, arXiv:0911.4975 [math.NT], 2009.
%H A001815 Simon Plouffe, <a href="/A000051/a000051_2.pdf">1031 Generating Functions</a>, Appendix to Thesis, Montreal, 1992.
%H A001815 Aaron Robertson, <a href="https://doi.org/10.46298/dmtcs.261">Permutations containing and avoiding 123 and 132 patterns</a>, Discrete Math. and Theoret. Computer Sci., Vol. 3, No. 4 (1999), pp. 151-154; <a href="http://arxiv.org/abs/math/9903169">arXiv preprint</a>, arXiv:math/9903169 [math.CO], 1999.
%H A001815 Aaron Robertson, <a href="http://arXiv.org/abs/math.CO/0012029">Permutations restricted by two distinct patterns of length three</a>, arXiv:math/0012029 [math.CO], 2000.
%H A001815 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (6,-12,8).
%F A001815 G.f.: 2*x^2/(1 - 2*x)^3. [_Simon Plouffe_ in his 1992 dissertation]
%F A001815 a(n) = A090802(n, 2).
%F A001815 a(n) = Sum_{i=0..n} i*(n-i)*binomial(n, i). - _Benoit Cloitre_, Nov 11 2004
%F A001815 a(n) = Sum_{k=0..n} k*2^(k-1). - _Zerinvary Lajos_, Oct 09 2006
%F A001815 a(n) = Sum_{j=0..n} binomial(n-1,j)*n*j. - _Zerinvary Lajos_, Oct 19 2006
%F A001815 E.g.f.: x^2*exp(2*x). - _Geoffrey Critzer_, Apr 07 2009
%F A001815 a(n) = 2^(n-2)*n*(n-1). - Tobias Friedrich (tfried(AT)mpi-inf.mpg.de), Jun 18 2009
%F A001815 a(n) = 2*a(n-1) + n*2^n.
%F A001815 For n > 0, a(n) = 2*A001788(n-1). - _Stanislav Sykora_, Oct 23 2016
%F A001815 a(n) = a(1-n) * 2^(2*n-1) for all n in Z. - _Michael Somos_, Oct 25 2016
%F A001815 a(n) = Sum_{k=0..n-1} Sum_{i=0..n-1} k * binomial(n-1,i). - _Wesley Ivan Hurt_, Sep 20 2017
%F A001815 From _Amiram Eldar_, Jan 07 2022: (Start)
%F A001815 Sum_{n>=0} 1/a(n) = 2*(1 - log(2)).
%F A001815 Sum_{n>=0} (-1)^n/a(n) = 6*log(3/2) - 2. (End)
%e A001815 G.f. = 2*x^2 + 12*x^3 + 48*x^4 + 160*x^5 + 480*x^6 + 1344*x^7 + 3584*x^8 + ...
%p A001815 A001815 := proc(n)
%p A001815     2^(n-2)*n*(n-1) ;
%p A001815 end proc: # _R. J. Mathar_, Mar 12 2014
%t A001815 Table[Binomial[n, 2]*2^(n-1), {n, 0, 28}] (* _Arkadiusz Wesolowski_, Dec 21 2011 *)
%t A001815 CoefficientList[Series[2 x^2/(1 - 2 x)^3, {x, 0, 40}], x] (* _Vincenzo Librandi_, Mar 14 2014 *)
%t A001815 LinearRecurrence[{6,-12,8},{0,0,2},30] (* _Harvey P. Dale_, May 19 2018 *)
%o A001815 (Sage) [lucas_number1(n, 2, 0)*binomial(n,2) for n in range(0, 29)] # _Zerinvary Lajos_, Mar 10 2009
%o A001815 (PARI) a(n)=binomial(n,2)<<(n-1) \\ _Charles R Greathouse IV_, Dec 21 2011
%o A001815 (PARI) my(x='x+O('x^100)); concat([0, 0], Vec(2*x^2/(1-2*x)^3)) \\ _Altug Alkan_, Nov 01 2015
%o A001815 (Magma) [Binomial(n,2)*2^(n-1): n in [0..30]]; // _Vincenzo Librandi_, Mar 14 2014
%Y A001815 Cf. A001788, A089264, A090802.
%K A001815 nonn,easy
%O A001815 0,3
%A A001815 _N. J. A. Sloane_

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