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A082582 Expansion of (1 + x^2 - sqrt( 1 - 4*x + 2*x^2 + x^4)) / (2*x) in powers of x. 60
1, 1, 1, 2, 5, 13, 35, 97, 275, 794, 2327, 6905, 20705, 62642, 190987, 586219, 1810011, 5617914, 17518463, 54857506, 172431935, 543861219, 1720737981, 5459867166, 17369553427, 55391735455, 177040109419, 567019562429, 1819536774089 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,4

COMMENTS

a(n) is the number of Dyck paths of semilength n with no UUDD. See A025242 for a bijection between paths avoiding DDUU versus UUDD.

Also number of lattice paths from (0,0) to (n,n) which do not go above the diagonal x=y using steps (1,k), (k,1) with k>=1. - Alois P. Heinz, Oct 07 2015

a(n) is the number of bargraphs of semiperimeter n (n>=2). Example: a(4) = 5; the 5 bargraphs correspond to the compositions [1,1,1], [1,2], [2,1], [2,2], [3]. - Emeric Deutsch, May 20 2016

a(n) is the number of skew Motzkin paths of length n. A skew Motzkin path is a path in the first quadrant which begins at the origin, ends on the x-axis, consists of steps U=(1,1) (up), D=(1,-1) (down), F=(1,0) (flat) and A=(-1,1) (anti-down) so that down and anti-down steps do not overlap. - Sergey Kirgizov, Oct 03 2018

LINKS

Reinhard Zumkeller, Table of n, a(n) for n = 0..1000

A. Blecher, C. Brennan, and A. Knopfmacher, Peaks in bargraphs, Trans. Royal Soc. South Africa, 71, No. 1, 2016, 97-103.

M. Bousquet-Mélou and A. Rechnitzer, The site-perimeter of bargraphs, Adv. in Appl. Math. 31 (2003), 86-112.

Emeric Deutsch, Sergi Elizalde, Statistics on bargraphs viewed as cornerless Motzkin paths, arXiv:1609.00088 [math.CO], September 2016.

Qing Lin Lu, Skew Motzkin paths, Acta Mathematica Sinica, English Series, 33(5) (2017), 657-667.

FORMULA

G.f.: (1 + x^2 - sqrt( 1 - 4*x + 2*x^2 + x^4)) / (2*x) = 2 /(1 + x^2 + sqrt( 1 - 4*x + 2*x^2 + x^4)).

G.f. A(x) satisfies the equation 0 = 1 - (1 + x^2) * A(x) + x * A(x)^2. - Michael Somos, Jul 22 2003

G.f. A(x) satisfies A(x) = 1 / (1 + x^2 - x * A(x)). - Michael Somos, Mar 28 2011

G.f. A(x) = 1 / (1 + x^2 - x / (1 + x^2 - x / (1 + x^2 - ... ))) continued fraction. - Michael Somos, Jul 01 2011

Series reversion of x * A(x) is x * A007477(-x). - Michael Somos, Jul 22 2003

a(n+1) = a(n) + Sum(a(k)*a(n-k): k=2..n), a(0) = a(1) = 1. - Reinhard Zumkeller, Nov 13 2012

G.f.: 1 + x - x*G(0), where G(k)= 1 - 1/(1 - x/(1 - x/(1 - x/(1 - x/(x - 1/G(k+1) ))))); (continued fraction). - Sergei N. Gladkovskii, Jul 12 2013

(n-1)*a(n)+(2*n+4)*a(n+2)+(-14-4*n)*a(n+3)+(5+n)*a(n+4) = 0. - Robert Israel, May 20 2016

EXAMPLE

1 + x + x^2 + 2*x^3 + 5*x^4 + 13*x^5 + 35*x^6 + 97*x^7 + 275*x^8 + ...

a(3)=2 because the only Dyck paths of semilength 3 with no UUDD in them are UDUDUD and UUDUDD (the nonqualifying ones being UDUUDD, UUDDUD and UUUDDD). - Emeric Deutsch, Jan 27 2003

MAPLE

f:= gfun:-rectoproc({(n-1)*a(n)+(2*n+4)*a(n+2)+(-14-4*n)*a(n+3)+(5+n)*a(n+4), a(0) = 1, a(1) = 1, a(2) = 1, a(3) = 2}, a(n), remember):

map(f, [$0..100]); # Robert Israel, May 20 2016

MATHEMATICA

a[0]=1; a[n_Integer]:=a[n]=a[n-1]+Sum[a[k]*a[n-1-k], {k, 2, n-1}]; Table[a[n], {n, 0, 40}] (* Vladimir Joseph Stephan Orlovsky, Mar 30 2011 *)

a[ n_] := SeriesCoefficient[ 2 / (1 + x^2 + Sqrt[1 - 4 x + 2 x^2 + x^4]), {x, 0, n}] (* Michael Somos, Jul 01 2011 *)

PROG

(PARI) {a(n) = polcoeff( (1 + x^2 - sqrt( 1 - 4*x + 2*x^2 + x^4 + x^2 * O(x^n))) / 2, n+1)} /* Michael Somos, Jul 22 2003 */

(PARI) {a(n) = if( n<0, 0, polcoeff( 2 /(1 + x^2 + sqrt( 1 - 4*x + 2*x^2 + x^4 + x * O(x^n))), n))} /* Michael Somos, Jul 01 2011 */

(PARI) {a(n) = local(A); if( n<0, 0, A = O(x); for( k = 0, n, A = 1 / (1 + x^2 - x * A)); polcoeff( A, n))} /* Michael Somos, Mar 28 2011 */

(Haskell)

a082582 n = a082582_list !! n

a082582_list = 1 : 1 : f [1, 1] where

   f xs'@(x:_:xs) = y : f (y : xs') where

     y = x + sum (zipWith (*) xs' $ reverse xs)

-- Reinhard Zumkeller, Nov 13 2012

CROSSREFS

Apart from initial term, same as A025242.

See A086581 for Dyck paths avoiding DDUU.

Cf. A218321, A263316.

Sequence in context: A007689 A085281 * A086581 A059027 A025198 A221205

Adjacent sequences:  A082579 A082580 A082581 * A082583 A082584 A082585

KEYWORD

easy,nonn

AUTHOR

Emanuele Munarini, May 07 2003

STATUS

approved

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Last modified November 20 02:52 EST 2018. Contains 317370 sequences. (Running on oeis4.)