%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 <a href="Sindx_Rea.html#recLCC">Index entries for sequences related to 
               linear recurrences with constant coefficients</a>
%H A001793 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 A001793 D. Callan, <a href="http://arXiv.org/abs/math.CO/0211380">A recursive 
               bijective approach to counting permutations...</a>
%H A001793 Milan Janjic, <a href="http://www.pmfbl.org/janjic/">Two Enumerative 
               Functions</a>
%H A001793 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 A001793 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 A001793 A. Robertson, H. S. Wilf and D. Zeilberger, Permutation patterns and 
               continued fractions, <a href="http://www.combinatorics.org/">Electr. 
               J. Combin.</a> 6, 1999, #R38.
%H A001793 <a href="Sindx_Ch.html#Cheby">Index entries for sequences related to 
               Chebyshev polynomials.</a>
%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)