Search: id:A001788 Results 1-1 of 1 results found. %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 Index entries for sequences related to linear recurrences with constant coefficients %H A001788 Milan Janjic, Two Enumerative Functions %H A001788 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 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. %H A001788 S. Plouffe, 1031 Generating Functions and Conjectures, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al, 1992. %H A001788 Eric Weisstein's World of Mathematics, Link to a section of The World of Mathematics. %H A001788 Eric Weisstein's World of Mathematics, Hypercube %H A001788 Index entries for sequences related to Chebyshev polynomials. %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 Search completed in 0.002 seconds