%I A001788 M4161 N1729
%S A001788 0,1,6,24,80,240,672,1792,4608,11520,28160,67584,159744,372736,
%T A001788 860160,1966080,4456448,10027008,22413312,49807360,110100480,
%U A001788 242221056,530579456,1157627904,2516582400,5452595200,11777605632
%N A001788 n*(n+1)*2^(n-2).
%C A001788 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
%C A001788 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.
%C A001788 Sum(i^2 * binomial(n, i), i=1..n) = 2^(n-2)*n*(n+1) - Yong Kong (ykong(AT)curagen.com), 
               Dec 26 2000
%C A001788 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
%C A001788 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
%C A001788 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
%C A001788 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
%C A001788 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
%C A001788 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]
%D A001788 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, 
               Academic Press, 1995 (includes this sequence).
%D A001788 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 
               (includes this sequence).
%D A001788 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.
%D A001788 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.
%D A001788 H. Izbicki, Ueber Unterbaeume eines Baumes, Monatshefte f\"{u}r Mathematik, 
               74 (1970), 56-62.
%D A001788 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.
%D A001788 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.
%D A001788 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.
%H A001788 <a href="Sindx_Rea.html#recLCC">Index entries for sequences related to 
               linear recurrences with constant coefficients</a>
%H A001788 Milan Janjic, <a href="http://www.pmfbl.org/janjic/">Two Enumerative 
               Functions</a>
%H A001788 M. Abramowitz and I. A. Stegun, eds., <a href="http://www.nrbook.com/
               abramowitz_and_stegun/">Handbook of Mathematical Functions</a>, National 
               Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 
               [alternative scanned copy].
%H A001788 S. Plouffe, <a href="http://www.lacim.uqam.ca/%7Eplouffe/articles/MasterThesis.pdf">
               Approximations de S\'{e}ries G\'{e}n\'{e}ratrices et Quelques Conjectures</
               a>, Dissertation, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al, 
               1992.
%H A001788 S. Plouffe, <a href="http://www.lacim.uqam.ca/%7Eplouffe/articles/FonctionsGeneratrices.pdf">
               1031 Generating Functions and Conjectures</a>, Universit\'{e} du 
               Qu\'{e}bec \`{a} Montr\'{e}al, 1992.
%H A001788 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
               IdempotentNumber.html">Link to a section of The World of Mathematics.</
               a>
%H A001788 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
               Hypercube.html">Hypercube</a>
%H A001788 <a href="Sindx_Ch.html#Cheby">Index entries for sequences related to 
               Chebyshev polynomials.</a>
%F A001788 G.f.: x/(1-2x)^3. E.g.f.: exp(2x)(x+x^2).
%F A001788 a(n) = sum(binomial(n+1,j)*(n+1-j)^2,j=0..n). - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), 
               Aug 22 2006
%F A001788 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
%F A001788 G.f.: x*(1-x)/exp(2*x) . [From Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), 
               Apr 03 2009]
%p A001788 A001788 := n->n*(n+1)*2^(n-2);
%p A001788 A001788:=-1/(2*z-1)**3; [S. Plouffe in his 1992 dissertation. Gives sequence 
               without initial zero.]
%p A001788 seq(seq(binomial(i+1, j)*2^(i-1), j =i-1), i=0..27); - Zerinvary Lajos 
               (zerinvarylajos(AT)yahoo.com), Dec 29 2007
%p A001788 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]
%o A001788 (PARI) a(n)=if(n<0,0,2^n*n*(n+1)/4)
%o A001788 (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]
%o A001788 (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]
%Y A001788 Cf. A001787, A001789.
%Y A001788 a(n)=2*a(n-1)+A001787(n-1). a(n)= A055252(n, 2).
%Y A001788 Row sums of triangle A094305.
%Y A001788 Cf. A001844.
%Y A001788 Cf. A038207.
%Y A001788 Sequence in context: A140088 A011855 A004404 this_sequence A068711 A047790 
               A133474
%Y A001788 Adjacent sequences: A001785 A001786 A001787 this_sequence A001789 A001790 
               A001791
%K A001788 nonn,easy,nice
%O A001788 0,3
%A A001788 N. J. A. Sloane (njas(AT)research.att.com).
%E A001788 Removed attribute "conjectured" from Plouffe g.f R. J. Mathar (mathar(AT)strw.leidenuniv.nl), 
               Mar 11 2009