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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], . - Emeric Deutsch, May 20 2016 [a(n) are the row sums of A271942 for n >= 2. Peter Luschny, Oct 18 2020] 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 From Gus Wiseman, Jul 04 2019: (Start) Conjecture: Also the number of maximal simple graphs with vertices {1..n} and no weakly nesting edges. Two edges {a,b}, {c,d} are weakly nesting if a <= c < d <= b or c <= a < b <= d. For example, the a(1) = 1 through a(5) = 13 edge-sets are:   {}  {12}  {13}     {14}        {15}             {12,23}  {12,24}     {12,25}                      {13,24}     {13,25}                      {13,34}     {14,25}                      {12,23,34}  {14,35}                                  {14,45}                                  {12,23,35}                                  {12,24,35}                                  {12,24,45}                                  {13,24,35}                                  {13,24,45}                                  {13,34,45}                                  {12,23,34,45} Cf. A006125, A054726, A117662, A326244, A326257, A326289, A326293, A326329, A326337, A326338, A326340. (End) a(n) is the number of Dyck n-paths in which no nonterminal descent has the same length as the preceding ascent. Example: a(3) = 2 counts UUDUDD and UUUDDD where the latter path qualifies because DDD is the terminal descent. - David Callan, Dec 14 2021 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. Miklos Bona and Elijah DeJonge, Pattern avoiding permutations and involutions with a unique longest increasing subsequence, arXiv:2003.10640 [math.CO], 2020. M. Bousquet-Mélou and A. Rechnitzer, The site-perimeter of bargraphs, Adv. in Appl. Math. 31 (2003), 86-112. Ralph L. Childress, Recursive Prime Factorizations: Dyck Words as Numbers, arXiv:2102.02777 [cs.FL], 2021. Emeric Deutsch and Sergi Elizalde, Statistics on bargraphs viewed as cornerless Motzkin paths, arXiv:1609.00088 [math.CO], September 2016. Chris Dyer, Gábor Melis, and Phil Blunsom, A Critical Analysis of Biased Parsers in Unsupervised Parsing, arXiv:1909.09428 [cs.CL], 2019. Juan B. Gil and Michael D. Weiner, On pattern-avoiding Fishburn permutations, arXiv:1812.01682 [math.CO], 2018-2019. 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 D-finite with recurrence: (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 a(n) = Sum_{k=0..n-2} Sum_{j=0..n-k-1} C(n-k-2,j)*C(k,j)*C(k+j+2,j)/(j+1), n>1, a(0)=1, a(1)=1. - Vladimir Kruchinin, Oct 18 2020 a(n) = Sum_{k=0..n-2} HypergeometricPFQ[{-k, 3 +k, k - n + 2}, {1, 2}, 1] for n >= 2. - Peter Luschny, Oct 18 2020 a(n) ~ sqrt(2+r) / (2 * sqrt(Pi) * n^(3/2) * r^n), where r = 0.295597742522084... is the real root of the equation r^3 + r^2 + 3*r - 1 = 0. - Vaclav Kotesovec, Jun 05 2022 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=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 *) a[n_] := Sum[HypergeometricPFQ[{-k, 3 + k, k - n}, {1, 2}, 1], {k, 0, n}]; Join[{1, 1}, Table[a[n], {n, 0, 26}]] (* Peter Luschny, Oct 18 2020 *) 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 (Maxima) a(n):=sum(sum((binomial(n-k-2, j)*binomial(k, j)*binomial(k+j+2, j))/(j+1), j, 0, n-k-1), k, 0, n-2); /* Vladimir Kruchinin, Oct 18 2020 */ CROSSREFS Apart from initial term, same as A025242. See A086581 for Dyck paths avoiding DDUU. Cf. A000108, A218321, A263316, A271942 (refinement). Sequence in context: A264228 A007689 A085281 * A086581 A059027 A025198 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 October 6 09:29 EDT 2022. Contains 357263 sequences. (Running on oeis4.)