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A001788 a(n) = n*(n+1)*2^(n-2).
(Formerly M4161 N1729)
75
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

Comment 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. [From 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), A00217((n-3)/2). - 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 R. 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. and A038207 formatted as a square array: 2.rows (0,1,2,3,4...) 1 6 24 80 240 672 1792 4608 - 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

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.

H. Izbicki, Über Unterbaeume eines Baumes, Monatshefte fur Mathematik, 74 (1970), 56-62.

Jones, C. W.; Miller, J. C. P.; Conn, J. F. C.; Pankhurst, R. C.; Tables of Chebyshev polynomials. Proc. Roy. Soc. Edinburgh. Sect. A. 62, (1946). 187-203.

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

T. D. Noe, Table of n, a(n) for n = 0..500

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].

H. J. Brothers, Pascal's Prism: Supplementary Material.

Milan Janjic, Two Enumerative Functions

M. Janjic and B. Petkovic, A Counting Function, arXiv 1301.4550 [math.CO], 2013.

Dusko Letic, Nenad Cakic, Branko Davidovic, Ivana Berkovic and Eleonora Desnica, Some certain properties of the generalized hypercubical functions, Advances in Difference Equations, 2011, 2011:60.

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.

Simon Plouffe, 1031 Generating Functions and Conjectures, Université du Québec à Montréal, 1992.

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., 1995, 35, 181-187.

Eric Weisstein's World of Mathematics, Idempotent Number

Eric Weisstein's World of Mathematics, Hypercube

Index entries for sequences related to Chebyshev polynomials.

Index entries for linear recurrences with constant coefficients, signature (6,-12,8).

FORMULA

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

E.g.f.: exp(2x)(x+x^2).

a(n) = 2*a(n-1) + n*2^(n-1) = 2*a(n-1) + A001787(n).

a(n) = A055252(n, 2).

a(n) = Sum(i^2 * binomial(n, i), i=1..n): binomial transform of A000290. - Yong Kong, Dec 26 2000

a(n) = sum(binomial(n+1,j)*(n+1-j)^2,j=0..n). - 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

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 - 2 x)^3, {x, 0, 26}], x]

Table[n*(n+1)*2^(n-2), {n, 0, 100}]

With[{nn=30}, Join[{0}, Times@@@Thread[{Accumulate[Range[nn]], 2^Range[0, nn-1]}]]] (* or *) 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) [lucas_number2(n, 2, 0)*binomial(n, 2)/2^1-lucas_number2(n, 2, 0)*binomial(n, 2)/2^2 for n in xrange(1, 28)] # Zerinvary Lajos, Mar 12 2009

(Sage) [lucas_number1(n, 2, 0)*binomial(n, 2)/2 for n in xrange(1, 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

CROSSREFS

Cf. A001787, A001789, A001844, A038207, A001793 (sum of all nodes of integer compositions, n included).

Row sums of triangle A094305.

Cf. A000079.

Sequence in context: A173031 A004404 A201189 * A068711 A047790 A133474

Adjacent sequences:  A001785 A001786 A001787 * A001789 A001790 A001791

KEYWORD

nonn,easy,nice

AUTHOR

N. J. A. Sloane

STATUS

approved

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Last modified March 29 18:39 EDT 2017. Contains 284273 sequences.