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 A055580 Björner-Welker sequence: 2^n*(n^2 + n + 2) - 1. 13

%I

%S 1,7,31,111,351,1023,2815,7423,18943,47103,114687,274431,647167,

%T 1507327,3473407,7929855,17956863,40370175,90177535,200278015,

%U 442499071,973078527,2130706431,4647288831,10099884031,21877489663

%N Björner-Welker sequence: 2^n*(n^2 + n + 2) - 1.

%C a(n) is the d=1 Betti number of the complement of '3-equal' arrangements in n dimensional real space, see Björner-Welker reference, Table I, pp. 308-309, column '1' with k=3 and Th. 5.2, pp. 297-298.

%C Binomial transform of [1/2, 2/3, 3/4, 4/5, ...] = 1/2, 7/6, 31/12, 111/20, 351/30, 1023/42, ..., where 2, 6, 12, 20, ... = A002378 (deleting the zero). - _Gary W. Adamson_, Apr 28 2005

%C Number of three-dimensional block structures associated with n joint systems in the construction of stable underground structures. - _Richard M. Green_, Jul 26 2011

%C Number of monotone mappings from the chain with three points to the complete binary tree of height n (n+1 levels). For example, the seven monotone mappings from the chain with three points (denoted 1,2,3, in order) to the complete binary tree with two levels (with a the root of the tree, and b, c the atoms) are: f(1)=f(2)=f(3)=a; f(1)=f(2)=a, f(3)=b; f(1)=f(2)=a, f(3)=c; f(1)=a, f(2)=f(3)=b; f(1)=a, f(2)=f(3)=c; f(1)=f(2)=f(3)=b; f(1)=f(2)=f(3)=c. - _Pietro Codara_, Mar 26 2015

%D H. Barcelo and S. Smith, The discrete fundamental group of the order complex of B_n, Abstract 1020-05-141, 1020th Meeting Amer. Math. Soc., Cincinatti, Ohio, Oct 21-22, 2006.

%H Vincenzo Librandi, <a href="/A055580/b055580.txt">Table of n, a(n) for n = 0..1000</a>

%H H. Barcelo and R. Laubenbacher, <a href="http://dx.doi.org/10.1016/j.disc.2004.03.016">Perspectives on A-homotopy theory and its applications</a>, Discr. Math., 298 (2005), 39-61.

%H H. Barcelo and S. Smith, <a href="http://arxiv.org/abs/0711.0915">The discrete fundamental group of the order complex of B_n</a>, arXiv:0711.0915 [math.CO], 2007.

%H A. Björner and V. Welker, <a href="http://dx.doi.org/10.1006/aima.1995.1012">The homology of "k-equal" manifolds and related partition lattices</a>, Adv. Math., 110 (1995), 277-313.

%H Harry Crane, <a href="http://stat.rutgers.edu/home/hcrane/Papers/ajc.pdf">Left-right arrangements, set partitions, and pattern avoidance</a>, Australasian Journal of Combinatorics, 61(1) (2015), 57-72.

%H G.G. Kocharyan and A.M. Kulyukin, <a href="http://dx.doi.org/10.1007/BF02335014">Construction of a three-dimensional block structure on the basis of jointed rock parameters estimating the stability of underground workings</a>, Soil Mech. Found. Eng., 31 (1994), 62-66.

%H A. F. Y. Zhao, <a href="http://www.emis.de/journals/JIS/VOL17/Zhao/zhao3.html">Pattern Popularity in Multiply Restricted Permutations</a>, Journal of Integer Sequences, 17 (2014), #14.10.3.

%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (7,-18,20,-8).

%F a(n) = A055252(n+3, 3).

%F a(n) = Sum_{j=0..n-1} a(j) + A045618(n), n >= 1.

%F G.f.: 1/((1-2*x)^3*(1-x)).

%F Partial sums of A001788 (without leading zero). - _Paul Barry_, Jun 26 2003

%F a(n) = A001788(n) - A000337(n). - _Jon Perry_, Dec 12 2003

%F a(n) = A119258(n+4,n). - _Reinhard Zumkeller_, May 11 2006

%F E.g.f.: 2*(1 + 2*x + 2*x^2)*exp(2*x) - exp(x). - _G. C. Greubel_, Oct 28 2016

%F a(n) = Sum_{k=0..n+1} Sum_{i=0..n+1} i^2 * C(k,i). - _Wesley Ivan Hurt_, Sep 21 2017

%t Table[ n*(n+1)*2^(n-2), {n, 0, 26}] // Accumulate // Rest (* _Jean-François Alcover_, Jul 09 2013, after _Paul Barry_ *)

%t LinearRecurrence[{7,-18,20,-8},{1,7,31,111},30] (* _Harvey P. Dale_, Nov 27 2014 *)

%o (MAGMA) [2^n*(n^2+n+2)-1: n in [0..35]]; // _Vincenzo Librandi_, Jul 28 2011

%o (PARI) a(n)=(n^2+n+2)<<n-1 \\ _Charles R Greathouse IV_, Jul 28 2011

%Y Fourth column of triangle A055252.

%Y Cf. A055252, A055249, A045618, A000337, A001788, A066185.

%K easy,nonn

%O 0,2

%A _Wolfdieter Lang_, May 26 2000; revised Feb 12 2001

%E Edited (for consistency with change of offset) by _M. F. Hasler_, Nov 03 2012

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Last modified November 13 02:24 EST 2018. Contains 317119 sequences. (Running on oeis4.)