0,3

Number of 2-dimensional faces in (n+1)-dimensional hypercube; also number of 4-cycles in the (n+1)-dimensional hypercube - Henry Bottomley (se16(AT)btinternet.com), Apr 14 2000

Comment from Philippe DELEHAM, Apr 28 2004: a(n) is the sum, over all non-empty subsets E of {1, 2, ..., n}, of all elements of E. E.g. a(3) = 24: the non-empty 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.

Sum(i^2 * binomial(n, i), i=1..n) = 2^(n-2)*n*(n+1) - Yong Kong (ykong(AT)curagen.com), Dec 26 2000

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 are 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 (zim.moreau.mann(AT)wanadoo.fr), 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 (PachaNambi(AT)yahoo.com), 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 (agnus(AT)blic.net), 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 (zerinvarylajos(AT)yahoo.com), Dec 29 2007

For n>0 where [0]={}, the empty set, and [n]={1,2,...n} a(n) is the number of ways to seperate [n-1] into three non-overlapping intervals (allowed to be empty) and then choose a subset from each interval. [From Geoffrey Critzer (critzer.geoffrey(AT)usd443), Feb 07 2009]

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

S. Plouffe, Approximations de S\'{e}ries G\'{e}n\'{e}ratrices et Quelques Conjectures, Dissertation, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al, 1992.

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, Ueber Unterbaeume eines Baumes, Monatshefte f\"{u}r 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.

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.

Tosic R., Masulovic D., Stojmenovic I., Brunvoll J., Cyvin B. N. and Cyvin S. J., Enumeration of polyhex hydrocarbons to h = 17, J. Chem. Inf. Comput. Sci., 1995, 35, 181-187.

Index entries for sequences related to linear recurrences with constant coefficients

Milan Janjic, Two Enumerative Functions

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].

S. Plouffe, Approximations de S\'{e}ries G\'{e}n\'{e}ratrices et Quelques Conjectures, Dissertation, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al, 1992.

S. Plouffe, 1031 Generating Functions and Conjectures, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al, 1992.

Eric Weisstein's World of Mathematics, Link to a section of The World of Mathematics.

Eric Weisstein's World of Mathematics, Hypercube

Index entries for sequences related to Chebyshev polynomials.

G.f.: x/(1-2x)^3. E.g.f.: exp(2x)(x+x^2).

a(n) = sum(binomial(n+1,j)*(n+1-j)^2,j=0..n). - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Aug 22 2006

Binomial transform of A001844: (1, 5, 13, 25, 41,...); = double binomial transform of [1, 4, 4, 0, 0, 0,...]. - Gary W. Adamson (qntmpkt(AT)yahoo.com), Sep 02 2007

G.f.: x*(1-x)/exp(2*x) . [From Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Apr 03 2009]

A001788 := n->n*(n+1)*2^(n-2);

A001788:=-1/(2*z-1)**3; [S. Plouffe in his 1992 dissertation. Gives sequence without initial zero.]

seq(seq(binomial(i+1, j)*2^(i-1), j =i-1), i=0..27); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Dec 29 2007

restart: G(x):=x*(1-x)/exp(2*x): f[0]:=G(x): for n from 1 to 26 do f[n]:=diff(f[n-1], x) od: x:=0: seq(abs(f[n]), n=0..26); # [From Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Apr 03 2009]

(PARI) a(n)=if(n<0, 0, 2^n*n*(n+1)/4)

(Other) 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)] # [From Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Mar 12 2009]

(Other) SAGE: [lucas_number1(n, 2, 0)*binomial(n, 2)/2 for n in xrange(1, 28)] [From Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Mar 10 2009]

Cf. A001787, A001789.

a(n)=2*a(n-1)+A001787(n-1). a(n)= A055252(n, 2).

Row sums of triangle A094305.

Cf. A001844.

Cf. A038207.

Sequence in context: A140088 A011855 A004404 this_sequence A068711 A047790 A133474

Adjacent sequences: A001785 A001786 A001787 this_sequence A001789 A001790 A001791

nonn,easy,nice

N. J. A. Sloane (njas(AT)research.att.com).

Removed attribute "conjectured" from Plouffe g.f R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Mar 11 2009