1,2

Coefficients of Chebyshev polynomials.

Number of 132-avoiding permutations of [n+3] containing exactly two 123 patterns. - Emeric Deutsch (deutsch(AT)duke.poly.edu), Jul 13 2001

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

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

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

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]

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).

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.

A. Denise and R. Simion, Two combinatorial statistics on Dyck paths, Discrete Math., 137, 1995, 155-176).

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.

Index entries for sequences related to linear recurrences with constant coefficients

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].

D. Callan, A recursive bijective approach to counting permutations...

Milan Janjic, Two Enumerative Functions

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.

A. Robertson, H. S. Wilf and D. Zeilberger, Permutation patterns and continued fractions, Electr. J. Combin. 6, 1999, #R38.

Index entries for sequences related to Chebyshev polynomials.

G.f.: x*(1-x)/(1-2*x)^3. Binomial transform of squares [1, 4, 9, ...].

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

With two leading zeros, binomial transform of quarter-squares A002620. - Paul Barry (pbarry(AT)wit.ie), May 27 2003

a(n)=sum{k=0..n+2, C(n+2, k)Floor(k^2/4) } - Paul Barry (pbarry(AT)wit.ie), May 27 2003

Sum{i=0..j, binomial(i+1, 2)*binomial(j, i)}. - Jon Perry (perry(AT)globalnet.co.uk), Feb 26 2004

With one leading zero, binomial transform of triangular numbers A000217 . - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Aug 02 2005

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

Left-shifted sequence is binomial transform of left-shifted squares (A000290). - Franklin T. Adams-Watters (FrankTAW(AT)Netscape.net), Nov 29 2006

Binomial transform of a(n)=n^2 offset 1. a(3)=18. [From Al Hakanson (hawkuu(AT)gmail.com), Jun 01 2009]

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.

A001793 := n*(n+3)*2^(n-3);

A001793:=(-1+z)/(2*z-1)**3; [S. Plouffe in his 1992 dissertation.]

a(n) = A039991(n+3, 4) = A055252(n, 1).

Cf. A058396.

Sequence in context: A011845 A099450 A145129 this_sequence A093374 A000745 A128553

Adjacent sequences: A001790 A001791 A001792 this_sequence A001794 A001795 A001796

easy,nonn

N. J. A. Sloane (njas(AT)research.att.com) and Simon Plouffe (simon.plouffe(AT)gmail.com)