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

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

Offset: 0

Author

Wolfdieter Lang, May 26 2000; revised Feb 12 2001

Keywords

• easy
• nonn

Comments

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.

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 three-dimensional 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 1020-05-141, 1020th Meeting Amer. Math. Soc., Cincinatti, Ohio, Oct 21-22, 2006.

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..1000
• Henry Adams, Samir Shukla, and Anurag Singh, Čech complexes of hypercube graphs, arXiv:2212.05871 [math.CO], 2022.
• H. Barcelo and R. Laubenbacher, Perspectives on A-homotopy theory and its applications, Discr. Math., 298 (2005), 39-61.
• 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 "k-equal" manifolds and related partition lattices, Adv. Math., 110 (1995), 277-313.
• Harry Crane, Left-right arrangements, set partitions, and pattern avoidance, Australasian Journal of Combinatorics, 61(1) (2015), 57-72.
• Robert Davis and Greg Simay, Further Combinatorics and Applications of Two-Toned Tilings, arXiv:2001.11089 [math.CO], 2020.
• G.G. Kocharyan and A.M. Kulyukin, Construction of a three-dimensional block structure on the basis of jointed rock parameters estimating the stability of underground workings, Soil Mech. Found. Eng., 31 (1994), 62-66.
• 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).

Crossrefs

Fourth column of triangle A055252.

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

Programs

Magma

[2^n*(n^2+n+2)-1: n in [0..35]]; // Vincenzo Librandi, Jul 28 2011

Mathematica

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

PARI

a(n)=(n^2+n+2)<Charles R Greathouse IV, Jul 28 2011

Formula

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

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

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

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

Extensions

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