

A055580


BjörnerWelker sequence: 2^n*(n^2 + n + 2)  1.


13



1, 7, 31, 111, 351, 1023, 2815, 7423, 18943, 47103, 114687, 274431, 647167, 1507327, 3473407, 7929855, 17956863, 40370175, 90177535, 200278015, 442499071, 973078527, 2130706431, 4647288831, 10099884031, 21877489663
(list;
graph;
refs;
listen;
history;
text;
internal format)



OFFSET

0,2


COMMENTS

a(n) is the d=1 Betti number of the complement of '3equal' arrangements in n dimensional real space, see BjörnerWelker reference, Table I, pp. 308309, column '1' with k=3 and Th. 5.2, pp. 297298.
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
Number of threedimensional block structures associated with n joint systems in the construction of stable underground structures.  Richard M. Green, Jul 26 2011
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


REFERENCES

H. Barcelo and S. Smith, The discrete fundamental group of the order complex of B_n, Abstract 102005141, 1020th Meeting Amer. Math. Soc., Cincinatti, Ohio, Oct 2122, 2006.


LINKS

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
H. Barcelo and R. Laubenbacher, Perspectives on Ahomotopy theory and its applications, Discr. Math., 298 (2005), 3961.
H. Barcelo and S. Smith, The discrete fundamental group of the order complex of B_n, arXiv:0711.0915 [math.CO], 2007.
A. Björner and V. Welker, The homology of "kequal" manifolds and related partition lattices, Adv. Math., 110 (1995), 277313.
Harry Crane, Leftright arrangements, set partitions, and pattern avoidance, Australasian Journal of Combinatorics, 61(1) (2015), 5772.
G.G. Kocharyan and A.M. Kulyukin, Construction of a threedimensional block structure on the basis of jointed rock parameters estimating the stability of underground workings, Soil Mech. Found. Eng., 31 (1994), 6266.
A. F. Y. Zhao, Pattern Popularity in Multiply Restricted Permutations, Journal of Integer Sequences, 17 (2014), #14.10.3.
Index entries for linear recurrences with constant coefficients, signature (7,18,20,8).


FORMULA

a(n) = A055252(n+3, 3).
a(n) = Sum_{j=0..n1} a(j) + A045618(n), n >= 1.
G.f.: 1/((12*x)^3*(1x)).
Partial sums of A001788 (without leading zero).  Paul Barry, Jun 26 2003
a(n) = A001788(n)  A000337(n).  Jon Perry, Dec 12 2003
a(n) = A119258(n+4,n).  Reinhard Zumkeller, May 11 2006
E.g.f.: 2*(1 + 2*x + 2*x^2)*exp(2*x)  exp(x).  G. C. Greubel, Oct 28 2016
a(n) = Sum_{k=0..n+1} Sum_{i=0..n+1} i^2 * C(k,i).  Wesley Ivan Hurt, Sep 21 2017


MATHEMATICA

Table[ n*(n+1)*2^(n2), {n, 0, 26}] // Accumulate // Rest (* JeanFrançois Alcover, Jul 09 2013, after Paul Barry *)
LinearRecurrence[{7, 18, 20, 8}, {1, 7, 31, 111}, 30] (* Harvey P. Dale, Nov 27 2014 *)


PROG

(MAGMA) [2^n*(n^2+n+2)1: n in [0..35]]; // Vincenzo Librandi, Jul 28 2011
(PARI) a(n)=(n^2+n+2)<<n1 \\ Charles R Greathouse IV, Jul 28 2011


CROSSREFS

Fourth column of triangle A055252.
Cf. A055252, A055249, A045618, A000337, A001788, A066185.
Sequence in context: A160607 A205492 A109756 * A097786 A197649 A006458
Adjacent sequences: A055577 A055578 A055579 * A055581 A055582 A055583


KEYWORD

easy,nonn


AUTHOR

Wolfdieter Lang, May 26 2000; revised Feb 12 2001


EXTENSIONS

Edited (for consistency with change of offset) by M. F. Hasler, Nov 03 2012


STATUS

approved



