Search: id:A001793 Results 1-1 of 1 results found. %I A001793 M3881 N1591 %S A001793 1,5,18,56,160,432,1120,2816,6912,16640,39424,92160,212992,487424, %T A001793 1105920,2490368,5570560,12386304,27394048,60293120,132120576,288358400, %U A001793 627048448,1358954496,2936012800,6325010432,13589544960,29125246976 %N A001793 a(n) = n*(n+3)*2^(n-3). %C A001793 Coefficients of Chebyshev polynomials. %C A001793 Number of 132-avoiding permutations of [n+3] containing exactly two 123 patterns. - Emeric Deutsch (deutsch(AT)duke.poly.edu), Jul 13 2001 %C A001793 Number of Dyck paths of semilength n+2 having pyramid weight n+1 (for pyramid weight see Denise and Simion). Example: a(2)=5 because the Dyck paths of semilength 4 having pyramid weight 3 are: (ud)u(ud)(ud)d, u(ud)(ud)d(ud), u(ud)(ud)(ud)d, u(ud)(uudd)d and u(uudd)(ud)d [here u=(1,1), d=(1,-1) and the maximal pyramids, of total length 3, are shown between parentheses]. - Emeric Deutsch (deutsch(AT)duke.poly.edu), Mar 10 2004 %C A001793 a(n) = number of dissections of a regular (n+3)-gon using n-1 noncrossing diagonals such that every piece of the dissection contains at least one non-base side of the (n+3)-gon. (One side of the (n+3)-gon is designated the base.) - David Callan (callan(AT)stat.wisc.edu), Mar 23 2004 %C A001793 If X_1,X_2,...,X_n are 2-blocks of a (2n+1)-set X then a(n) is the number of (n+2)-subsets of X intersecting each X_i, (i=1,2,...,n). - Milan R. Janjic (agnus(AT)blic.net), Nov 18 2007 %C A001793 The second corrector line for transforming 2^n offset 0 with a leading 1 into the fibonacci sequence. [From Al Hakanson (hawkuu(AT)gmail.com), Jun 01 2009] %D A001793 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence). %D A001793 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence). %D A001793 M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 795. %D A001793 A. Denise and R. Simion, Two combinatorial statistics on Dyck paths, Discrete Math., 137, 1995, 155-176). %D A001793 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. %H A001793 Index entries for sequences related to linear recurrences with constant coefficients %H A001793 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 A001793 D. Callan, A recursive bijective approach to counting permutations... %H A001793 Milan Janjic, Two Enumerative Functions %H A001793 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 A001793 S. Plouffe, 1031 Generating Functions and Conjectures, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al, 1992. %H A001793 A. Robertson, H. S. Wilf and D. Zeilberger, Permutation patterns and continued fractions, Electr. J. Combin. 6, 1999, #R38. %H A001793 Index entries for sequences related to Chebyshev polynomials. %F A001793 G.f.: x*(1-x)/(1-2*x)^3. Binomial transform of squares [1, 4, 9, ...]. %F A001793 a(n)=sum{k=0..floor((n+4)/2), C(n+4, 2k)C(k, 2) } - Paul Barry (pbarry(AT)wit.ie), May 15 2003 %F A001793 With two leading zeros, binomial transform of quarter-squares A002620. - Paul Barry (pbarry(AT)wit.ie), May 27 2003 %F A001793 a(n)=sum{k=0..n+2, C(n+2, k)Floor(k^2/4) } - Paul Barry (pbarry(AT)wit.ie), May 27 2003 %F A001793 Sum{i=0..j, binomial(i+1, 2)*binomial(j, i)}. - Jon Perry (perry(AT)globalnet.co.uk), Feb 26 2004 %F A001793 With one leading zero, binomial transform of triangular numbers A000217 . - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Aug 02 2005 %F A001793 a(n)=sum{k=0..n+1, (-1)^(n-k+1)C(k, n-k+1)*k*C(2k, k)/2}; - Paul Barry (pbarry(AT)wit.ie), Oct 07 2005 %F A001793 Left-shifted sequence is binomial transform of left-shifted squares (A000290). - Franklin T. Adams-Watters (FrankTAW(AT)Netscape.net), Nov 29 2006 %F A001793 Binomial transform of a(n)=n^2 offset 1. a(3)=18. [From Al Hakanson (hawkuu(AT)gmail.com), Jun 01 2009] %e A001793 a(2)=5 since 32415, 32451, 34125, 42135 and 52134 are the only 132-avoiding permutations of 12345 containing exactly two increasing subsequences of length 3. %p A001793 A001793 := n*(n+3)*2^(n-3); %p A001793 A001793:=(-1+z)/(2*z-1)**3; [S. Plouffe in his 1992 dissertation.] %Y A001793 a(n) = A039991(n+3, 4) = A055252(n, 1). %Y A001793 Cf. A058396. %Y A001793 Sequence in context: A011845 A099450 A145129 this_sequence A093374 A000745 A128553 %Y A001793 Adjacent sequences: A001790 A001791 A001792 this_sequence A001794 A001795 A001796 %K A001793 easy,nonn %O A001793 1,2 %A A001793 N. J. A. Sloane (njas(AT)research.att.com) and Simon Plouffe (simon.plouffe(AT)gmail.com) Search completed in 0.002 seconds