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A001788 a(n) = n*(n+1)*2^(n-2).
(Formerly M4161 N1729)
105
0, 1, 6, 24, 80, 240, 672, 1792, 4608, 11520, 28160, 67584, 159744, 372736, 860160, 1966080, 4456448, 10027008, 22413312, 49807360, 110100480, 242221056, 530579456, 1157627904, 2516582400, 5452595200, 11777605632 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,3
COMMENTS
Number of 2-dimensional faces in (n+1)-dimensional hypercube; also number of 4-cycles in the (n+1)-dimensional hypercube. - Henry Bottomley, Apr 14 2000
Also the number of edges in the (n+1)-halved cube graph. - Eric W. Weisstein, Jun 21 2017
From Philippe Deléham, Apr 28 2004: a(n) is the sum, over all nonempty subsets E of {1, 2, ..., n}, of all elements of E. E.g., a(3) = 24: the nonempty subsets are {1, 2, 3}, {1, 2}, {1, 3}, {2, 3}, {1}, {2}, {3} and 1 + 2 + 3 + 1 + 2 + 1 + 3 + 2 + 3 + 1 + 2 + 3 = 24.
Equivalently, sum of all nodes (except the last one, equal to n+1) of all integer compositions of n+1. - Olivier Gérard, Oct 22 2011
The inverse binomial transform of a(n-k) for k=-1..4 gives A001844, A000290, A000217(n-1), A002620(n-1), A008805(n-4), A000217 interspersed with 0's. - Michael Somos, Jul 18 2003
Take n points on a finite line. They all move with the same constant speed; they instantaneously change direction when they collide with another; and they fall when they quit the line. a(n-1) is the total number of collisions before falling when the initials directions are the 2^n possible. The mean number of collisions is then n(n-1)/8. E.g., a(1)=0 before any collision is possible. a(2)=1 because there is a collision only if the initials directions are, say, right-left. - Emmanuel Moreau, Feb 11 2006
Also number of pericondensed hexagonal systems with n hexagons. For example, if n=5 then the number of pericondensed hexagonal systems with n hexagons is 24. - Parthasarathy Nambi, Sep 06 2006
If X_1,X_2,...,X_n is a partition of a 2n-set X into 2-blocks then, for n>1, a(n-1) is equal to the number of (n+2)-subsets of X intersecting each X_i (i=1,2,...,n). - Milan Janjic, Jul 21 2007
Number of n-permutations of 3 objects u,v,w, with repetition allowed, containing exactly two u's. Example: a(2)=6 because we have uuw, uuv, uwu, uvu, wuu and vuu. - Zerinvary Lajos, Dec 29 2007
For n>0 where [0]={}, the empty set, and [n]={1,2,...n} a(n) is the number of ways to separate [n-1] into three non-overlapping intervals (allowed to be empty) and then choose a subset from each interval. - Geoffrey Critzer, Feb 07 2009
Form an array with m(n,0) = m(0,n) = n^2 and m(i,j) = m(i-1,j-1) + m(i-1,j). Then m(1,n) = A001844(n) and m(n,n) = a(n). - J. M. Bergot, Nov 07 2012
The sum of the number of inversions of all sequences of zeros and ones with length n+1. - Evan M. Bailey, Dec 09 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. 796.
Clifford A. Pickover, The Math Book, From Pythagoras to the 57th Dimension, 250 Milestones in the History of Mathematics, Sterling Publ., NY, 2009, page 282.
A. P. Prudnikov, Yu. A. Brychkov and O.I. Marichev, "Integrals and Series", Volume 1: "Elementary Functions", Chapter 4: "Finite Sums", New York, Gordon and Breach Science Publishers, 1986-1992.
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
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].
Geoffrey Critzer, Combinatorics of Vector Spaces over Finite Fields, Master's thesis, Emporia State University, 2018.
Robert Davis and Greg Simay, Further Combinatorics and Applications of Two-Toned Tilings, arXiv:2001.11089 [math.CO], 2020.
Herbert Izbicki, Über Unterbäume eines Baumes, Monatshefte fur Mathematik, Vol. 74 (1970), pp. 56-62.
Milan Janjic and Boris Petkovic, A Counting Function, arXiv 1301.4550 [math.CO], 2013.
C. W. Jones, J. C. P. Miller, J. F. C. Conn, and R. C. Pankhurst, Tables of Chebyshev polynomials, Proc. Roy. Soc. Edinburgh. Sect. A., Vol. 62, No. 2 (1946), pp. 187-203.
Han Mao Kiah, Alexander Vardy, and Hanwen Yao, Efficient Algorithms for the Bee-Identification Problem, arXiv:2212.09952 [cs.IT], 2022.
Duško Letić, Nenad Cakić, Branko Davidović, Ivana Berković and Eleonora Desnica, Some certain properties of the generalized hypercubical functions, Advances in Difference Equations, Vol. 2011 (2011), Article 60.
Mircea Merca, A Special Case of the Generalized Girard-Waring Formula, J. Integer Sequences, Vol. 15 (2012), Article 12.5.7.
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.
Lara Pudwell, Nathan Chenette and Manda Riehl, Statistics on Hypercube Orientations, AMS Special Session on Experimental and Computer Assisted Mathematics, Joint Mathematics Meetings (Denver 2020).
John Riordan and N. J. A. Sloane, Correspondence, 1974.
R. Tosic, D. Masulovic, I. Stojmenovic, J. Brunvoll, B. N. Cyvin and S. J. Cyvin, Enumeration of polyhex hydrocarbons to h = 17, J. Chem. Inf. Comput. Sci., Vol. 35, No. 2 (1995), pp. 181-187.
Eric Weisstein's World of Mathematics, Edge Count.
Eric Weisstein's World of Mathematics, Graph Cycle.
Eric Weisstein's World of Mathematics, Idempotent Number.
Eric Weisstein's World of Mathematics, Halved Cube Graph.
Eric Weisstein's World of Mathematics, Hypercube Graph.
FORMULA
G.f.: x/(1-2*x)^3.
E.g.f.: x*(1 + x)*exp(2*x).
a(n) = 2*a(n-1) + n*2^(n-1) = 2*a(n-1) + A001787(n).
a(n) = A038207(n+1,2).
a(n) = A055252(n, 2).
a(n) = Sum_{i=1..n} i^2 * binomial(n, i): binomial transform of A000290. - Yong Kong, Dec 26 2000
a(n) = Sum_{j=0..n} binomial(n+1,j)*(n+1-j)^2. - Zerinvary Lajos, Aug 22 2006
If the leading 0 is deleted, the binomial transform of A001844: (1, 5, 13, 25, 41, ...); = double binomial transform of [1, 4, 4, 0, 0, 0, ...]. - Gary W. Adamson, Sep 02 2007
a(n) = Sum_{1<=i<=k<=n} (-1)^(i+1)*i^2*binomial(n+1,k+i)*binomial(n+1,k-i). - Mircea Merca, Apr 09 2012
a(0)=0, a(1)=1, a(2)=6, a(n) = 6*a(n-1) - 12*a(n-2) + 8*a(n-3). - Harvey P. Dale, Jul 16 2013
a(n) = Sum_{k=0..n-1} Sum_{i=0..n-1} (k+1) * C(n-1,i). - Wesley Ivan Hurt, Sep 20 2017
From Amiram Eldar, Jan 05 2022: (Start)
Sum_{n>=1} 1/a(n) = 4*(1-log(2)).
Sum_{n>=1} (-1)^(n+1)/a(n) = 12*log(3/2) - 4. (End)
EXAMPLE
The nodes of an integer composition are the partial sums of its elements, seen as relative distances between nodes of a 1-dimensional polygon. For a composition of 7 such as 1+2+1+3, the nodes are 0,1,3,4,7. Their sum (without the last node) is 8. The sum of all nodes of all 2^(7-1)=64 integer compositions of 7 is 672.
MAPLE
A001788 := n->n*(n+1)*2^(n-2);
A001788:=-1/(2*z-1)**3; # Simon Plouffe in his 1992 dissertation; gives sequence without initial zero
MATHEMATICA
CoefficientList[Series[x/(1-2x)^3, {x, 0, 30}], x]
Table[n*(n+1)*2^(n-2), {n, 0, 30}]
With[{n = 30}, Join[{0}, Times @@@ Thread[{Accumulate[Range[n]], 2^Range[0, n - 1]}]]] (* Harvey P. Dale, Jul 16 2013 *)
LinearRecurrence[{6, -12, 8}, {0, 1, 6}, 30] (* Harvey P. Dale, Jul 16 2013 *)
PROG
(PARI) a(n)=if(n<0, 0, 2^n*n*(n+1)/4)
(Sage) [n if n < 2 else n * (n + 1) * 2**(n - 2) for n in range(28)] # Zerinvary Lajos, Mar 10 2009
(Haskell)
a001788 n = if n < 2 then n else n * (n + 1) * 2 ^ (n - 2)
a001788_list = zipWith (*) a000217_list $ 1 : a000079_list
-- Reinhard Zumkeller, Jul 11 2014
(Magma) [n*(n+1)*2^(n-2): n in [0..30]]; // G. C. Greubel, Aug 27 2019
(GAP) List([0..30], n-> n*(n+1)*2^(n-2)); # G. C. Greubel, Aug 27 2019
CROSSREFS
Cf. A000079, A001787, A001789, A001793 (sum of all nodes of integer compositions, n included).
Cf. A001844, A038207, A290031 (6-cycles).
Row sums of triangle A094305.
Sequences similar to the form q^(n-2)*binomial(n, 2): A000217 (q=1), this sequence (q=2), A027472 (q=3), A038845 (q=4), A081135 (q=5), A081136 (q=6), A027474 (q=7), A081138 (q=8), A081139 (q=9), A081140 (q=10), A081141 (q=11), A081142 (q=12), A027476 (q=15).
Sequence in context: A004404 A319552 A201189 * A068711 A344273 A320856
KEYWORD
nonn,easy,nice
AUTHOR
STATUS
approved

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Last modified February 29 16:18 EST 2024. Contains 370425 sequences. (Running on oeis4.)