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A001793 a(n) = n*(n+3)*2^(n-3).
(Formerly M3881 N1591)
41
1, 5, 18, 56, 160, 432, 1120, 2816, 6912, 16640, 39424, 92160, 212992, 487424, 1105920, 2490368, 5570560, 12386304, 27394048, 60293120, 132120576, 288358400, 627048448, 1358954496, 2936012800, 6325010432, 13589544960, 29125246976 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

Coefficients of Chebyshev polynomials; a sub-diagonal in A053120.

Number of 132-avoiding permutations of [n+3] containing exactly two 123 patterns. - Emeric Deutsch, 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, 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, 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..n). - Milan Janjic, Nov 18 2007

The second corrector line for transforming 2^n offset 0 with a leading 1 into the fibonacci sequence. [Al Hakanson (hawkuu(AT)gmail.com), Jun 01 2009]

Sum of all nodes of all integer compositions of n, see example. - Olivier Gérard, Oct 22 2011

Number of compositions of 2n with exactly two odd summands, see example. - Mamuka Jibladze, Sep 04 2013

4*a(n) is the number of North-East paths from (0,0) to (n+2,n+2) that whose total number of east steps below y = x-1 or above y = x+1 is exactly two. It is related to paired pattern P_1 and P_6 in Pan and Remmel's link. - Ran Pan, Feb 04 2016

REFERENCES

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.

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

LINKS

T. D. Noe, Table of n, a(n) for n = 1..500

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 containing 3-letter patterns, arXiv:math/0211380 [math.CO], 2002.

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

Filippo Disanto, Some Statistics on the Hypercubes of Catalan Permutations, Journal of Integer Sequences, Vol. 18 (2015), Article 15.2.2.

Milan Janjic, Two Enumerative Functions

C. W. Jones, J. C. P. Miller, J. F. C. Conn, R. C. Pankhurst, Tables of Chebyshev polynomials Proc. Roy. Soc. Edinburgh. Sect. A. 62, (1946). 187-203.

Ran Pan, Jeffrey B. Remmel, Paired patterns in lattice paths, arXiv:1601.07988 [math.CO], 2016.

Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992.

Simon Plouffe, 1031 Generating Functions and Conjectures, Université du Québec à Montréal, 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.

Index entries for linear recurrences with constant coefficients, signature (6,-12,8).

FORMULA

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, May 15 2003

With two leading zeros, binomial transform of quarter-squares A002620. - Paul Barry, May 27 2003

a(n) = Sum_{k=0..n+2} C(n+2, k) * floor(k^2/4). - Paul Barry, May 27 2003

a(n) = Sum_{i=0..j} binomial(i+1, 2)*binomial(j, i). - Jon Perry, Feb 26 2004

With one leading zero, binomial transform of triangular numbers A000217. - Philippe Deléham, 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, Oct 07 2005

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

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

a(n) = (1/n) * Sum_{k=0..n} binomial(n,k)*k^3. - Gary Detlefs, Nov 26 2011

For n > 1, a(n) = Sum_{k=0..n-1} Sum_{i=0..n} (k+2) * C(n-2,i). - Wesley Ivan Hurt, Sep 20 2017

EXAMPLE

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.

a(4)=56: the compositions of 4 are 4, 3+1, 1+3, 2+2, 2+1+1, 1+2+1, 1+1+2, 1+1+1+1, the corresponding nodes (partial sums) are {0, 4}, {0, 3, 4}, {0, 1, 4}, {0, 2, 4}, {0, 2, 3, 4}, {0, 1, 3, 4}, {0, 1, 2, 4}, {0, 1, 2, 3, 4}, with individual sums {4, 7, 5, 6, 9, 8, 7, 10} and total 56. - Olivier Gérard, Oct 22 2011

The a(3)=18 compositions of 2*3=6 with two odd summands are 5+1, 1+5, 3+3, 4+1+1, 1+4+1, 1+1+4, 3+2+1, 3+1+2, 2+3+1, 2+1+3, 1+3+2, 1+2+3, 2+2+1+1, 2+1+2+1, 2+1+1+2, 1+2+2+1, 1+2+1+2, 1+1+2+2. - Mamuka Jibladze, Sep 04 2013

MAPLE

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

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

MATHEMATICA

Table[n (n + 3)*2^(n - 3), {n, 28}] (* or *)

Rest@ CoefficientList[Series[x (1 - x)/(1 - 2 x)^3, {x, 0, 28}], x] (* Michael De Vlieger, Sep 21 2017 *)

PROG

(PARI) a(n)=n*(n+3)<<(n-3) \\ Charles R Greathouse IV, Sep 24 2015

CROSSREFS

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

Cf. A058396.

Sequence in context: A011845 A099450 A145129 * A093374 A258109 A000745

Adjacent sequences:  A001790 A001791 A001792 * A001794 A001795 A001796

KEYWORD

easy,nonn,changed

AUTHOR

N. J. A. Sloane and Simon Plouffe

STATUS

approved

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Last modified September 26 06:54 EDT 2017. Contains 292502 sequences.