A263316 -id:A263316 - cp's OEIS frontend

A082582 Expansion of (1 + x^2 - sqrt( 1 - 4x + 2x^2 + x^4)) / (2*x) in powers of x.

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

Offset: 0

Emanuele Munarini, May 07 2003

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) 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

			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

Reinhard Zumkeller, Table of n, a(n) for n = 0..1000
Jean Luc Baril, Rigoberto Flórez, and José L. Ramirez, Generalized Narayana arrays, restricted Dyck paths, and related bijections, Univ. Bourgogne (France, 2025). See p. 11.
Jean-Luc Baril and José Luis Ramírez, Descent distribution on Catalan words avoiding ordered pairs of Relations, arXiv:2302.12741 [math.CO], 2023. See pp. 3, 13.
Aubrey Blecher, Charlotte Brennan, and Arnold 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.
Mireille Bousquet-Mélou and Andrew 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.
Toufik Mansour and Mark Shattuck, On ascent sequences avoiding 021 and a pattern of length four, arXiv:2507.17947 [math.CO], 2025. See p. 19.
Helmut Prodinger, Cornerless, peakless, valleyless Motzkin paths (regular and skew) and applications to bargraphs, arXiv:2501.13645 [math.CO], 2025. See p. 11.
Robin D. P. Zhou, Pattern avoidance in revised ascent sequences, arXiv:2505.05171 [math.CO], 2025. See pp. 4, 22.

Apart from initial term, same as A025242.
See A086581 for Dyck paths avoiding DDUU.
Cf. A000108, A218321, A263316, A271942 (refinement).
Column k=0 of A098978.

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

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

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 *)
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 *)

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 */

{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 */

{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 */

{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 */

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
G.f.: 1/G(x), with G(x) = 1 - (x-x^2)/(1-x/G(x)) (continued fraction). - Nikolaos Pantelidis, Jan 11 2023

A168592 G.f.: exp( Sum_{n>=1} A082758(n)*x^n/n ), where A082758(n) = sum of the squares of the trinomial coefficients in row n of triangle A027907.

Original entry on oeis.org

1, 3, 14, 80, 509, 3459, 24579, 180389, 1356743, 10402493, 81004516, 638886082, 5093081983, 40971735401, 332187974718, 2711668091448, 22267979870143, 183830653156341, 1524747465249750, 12700172705956876, 106187411693668179

Offset: 0

Paul D. Hanna, Dec 01 2009

nonn

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 >= 0 and two kinds of (1,1). - Alois P. Heinz, Oct 07 2015

Number of pairs of noncrossing paths of length n which start and end together, each taking steps (1,0), (1,1) or (1,-1) (i.e., Motzkin-type). - Nicholas R. Beaton, Jun 17 2024

			G.f.: A(x) = 1 + 3*x + 14*x^2 + 80*x^3 + 509*x^4 + 3459*x^5 + ...
log(A(x)) = 3*x + 19*x^2/2 + 141*x^3/3 + 1107*x^4/4 + 8953*x^5/5 + ... + A082758(n)*x^n/n + ...

Alois P. Heinz, Table of n, a(n) for n = 0..1000

Cf. A168590, A168593, A082758, A027907, A168595, A218321, A263316.

b:= proc(x, y) option remember; `if`(y<0 or y>x, 0, `if`(x=0, 1,
      add(b(x-i, y-1), i=0..x) +add(b(x-1, y-j), j=0..y)))
    end:
a:= n-> b(n$2):
seq(a(n), n=0..25);  # Alois P. Heinz, Oct 07 2015
# second Maple program:
a:= proc(n) option remember; `if`(n<4, [1, 3, 14, 80][n+1],
      ((10*(n+1))*(16*n^3-20*n^2-n-1) *a(n-1)
      +(-944*n^4+2596*n^3-1924*n^2+236*n+30) *a(n-2)
      +(90*(n-2))*(16*n^3-52*n^2+45*n-6) *a(n-3)
      -(81*(2*n-5))*(n-2)*(n-3)*(4*n-1) *a(n-4))/
      ((n+1)*(4*n-5)*(2*n+1)*(n+2)))
    end:
seq(a(n), n=0..25);  # Alois P. Heinz, Oct 07 2015

(1/x)*InverseSeries[x*(1 - x)^2/((1 + x)^2*(1 - x + x^2)) + O[x]^30, x] // CoefficientList[#, x]& (* Jean-François Alcover, Jun 09 2018 *)

{a(n)=if(n==0,1,polcoeff(exp(sum(m=1,n,sum(k=0,2*m, polcoeff((1+x+x^2)^m,k)^2)*x^m/m) +x*O(x^n)),n))}

{a(n)=polcoeff(1/x*serreverse(x*(1-x)^2/((1+x)^2*(1-x+x^2)+x*O(x^n))),n)}

G.f.: A(x) = (1/x)*Series_Reversion[x*(1-x)^2/((1+x)^2*(1-x+x^2))].
G.f.: A(x) satisfies A(x^2) = M(x)*M(-x), where M(x) is the g.f. of A001006. - Alexander Burstein, Oct 03 2017
G.f.: A(x) satisfies A(x^2) = (1-x - sqrt(1-2*x-3*x^2))*(1+x - sqrt(1+2*x-3*x^2))/(4*x^4). - Paul D. Hanna, Oct 05 2017, concluded from formula of Alexander Burstein.

A218321 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>=0.

Original entry on oeis.org

1, 2, 8, 39, 212, 1230, 7458, 46689, 299463, 1957723, 12996879, 87383754, 593794311, 4071599216, 28136612051, 195756911831, 1370068168916, 9639404836227, 68138551870047, 483682445360748, 3446462104490724, 24642148415136556, 176743014104068411

Offset: 0

Alois P. Heinz, Oct 25 2012

nonn

			a(2) = 8: [(0,0),(1,0),(1,1),(2,1),(2,2)], [(0,0),(1,0),(1,1),(2,2)], [(0,0),(1,0),(2,0),(2,1),(2,2)], [(0,0),(1,0),(2,1),(2,2)], [(0,0),(1,0),(2,2)], [(0,0),(1,1),(2,1),(2,2)], [(0,0),(1,1),(2,2)], [(0,0),(2,1),(2,2)].

Alois P. Heinz, Table of n, a(n) for n = 0..500
Alois P. Heinz, Maple program for A218321

Cf. A082582, A168592, A263316.

b:= proc(x, y) option remember; `if`(y<0 or y>x, 0, `if`(x=0, 1,
      add(b(x-i, y-1), i=0..x) +add(b(x-1, y-j), j=0..y) -b(x-1,y-1)))
    end:
a:= n-> b(n, n):
seq(a(n), n=0..30);
# second Maple program gives series:
series(RootOf(x^4*T^4-(x^2+1)*x^2*T^3-(x^2-2*x-2)*x*T^2-(x^2+1)*T+1, T), x=0, 31);  # Mark van Hoeij, Apr 17 2013

b[x_, y_] := b[x, y] = If[y < 0 || y > x, 0, If[x == 0, 1, Sum[b[x - i, y - 1], {i, 0, x}] + Sum[b[x - 1, y - j], {j, 0, y}] - b[x - 1, y - 1]]];
a[n_] := b[n, n];
Table[a[n], {n, 0, 30}] (* Jean-François Alcover, Sep 01 2022, after Alois P. Heinz *)

G.f.: (sqrt(x^4+4*x^3+2*x^2-8*x+1)+x^2+1-sqrt(2*(x^4+2*x^3-6*x^2-4*x+1+(x^2+1)*sqrt(x^4+4*x^3+2*x^2-8*x+1))))/(4*x^2). - Mark van Hoeij, Apr 17 2013

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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.

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A168592 G.f.: exp( Sum_{n>=1} A082758(n)*x^n/n ), where A082758(n) = sum of the squares of the trinomial coefficients in row n of triangle A027907.

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A218321 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>=0.

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A082582 Expansion of (1 + x^2 - sqrt( 1 - 4x + 2x^2 + x^4)) / (2*x) in powers of x.