A007477 -id:A007477 - cp's OEIS frontend

A213705 a(n)=n if n <= 3, otherwise a(n) = A007477(n-1) + A007477(n).

1, 2, 3, 5, 9, 17, 33, 66, 134, 277, 579, 1224, 2610, 5609, 12135, 26408, 57770, 126962, 280192, 620674, 1379586, 3075943, 6877611, 15417934, 34646156, 78027146, 176087292, 398143230, 901827322, 2046112299, 4649558191, 10581041518, 24112473412, 55019560650, 125696393844, 287494670302

Offset: 1

Antti Karttunen, Sep 14 2012

a(n) gives the number of "plausible parsings" of the sentence "Etsivät^(n+1)" in Finnish (with the most common word order, SV & SVO), that is, sentences which consist only of n+1 copies of the word "etsivät". See the OEIS Wiki page.
See A007477 for the number of plausible parsings of "Buffalo^n" sentences in English.
In my view the value of a(0) should be 0 in this context (single word "Etsivät." is not a valid Finnish sentence, except as an answer to a question), although this is arguable. However, it is probably that this sequence occurs in other (combinatorial) contexts as well, and there a(0) might be something else than 0, so I left it off, and made the sequence start from offset 1.

			G.f. = x + 2*x^2 + 3*x^3 + 5*x^4 + 9*x^5 + 17*x^6 + 33*x^7 + ... - _Michael Somos_, Nov 07 2019

Antti Karttunen, Table of n, a(n) for n = 1..1000
Antti Karttunen, Etsivät etsivät etsivät..., OEIS Wiki.

Cf. A007477, A007853.

b:= n-> coeff(series(RootOf(A=(A*x)^2+x+1, A), x, n+1), x, n):
a:= n-> `if`(n<2, n, b(n-1) +b(n)):
seq(a(n), n=1..40);  # Alois P. Heinz, Sep 14 2012

(* b = A007477 *) b[n_] := Sum[Binomial[2*k+2, n-k-2]*Binomial[n-k-2, k]/(k + 1), {k, 0, n-2}]; a[n_] := b[n-1] + b[n]; a[1] = 1; a[2] = 2; Array[a, 40] (* Jean-François Alcover, Mar 04 2016 *)

b(n) = sum(k=0, n - 2, binomial(2*k + 2, n - k - 2)*binomial(n - k - 2, k)/(k + 1));
a(n) = if(n<3, n, b(n - 1) + b(n)); \\ Indranil Ghosh, Apr 11 2017

{a(n) = polcoeff( (1 + x) * (1 - 2*x^2 - sqrt(1 - 4*x^2 - 4*x^3 + x^3 * O(x^n))) / (2*x^2), n)}; /* Michael Somos, Nov 07 2019 */

from sympy import binomial
def b(n): return sum([binomial(2*k + 2, n - k - 2)*binomial(n - k - 2, k)//(k + 1) for k in range(n - 1)])
def a(n): return n if n<3 else b(n - 1) + b(n)
print([a(n) for n in range(1, 51)]) # Indranil Ghosh, Apr 11 2017

: (define (A213705 n) (if (< n 2) n (+ (A007477 (- n 1)) (A007477 n))))

Given the g.f. A(x) and the g.f. of A007853 B(x), then -x = A(-B(x)). - Michael Somos, Nov 07 2019

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

Original entry on oeis.org

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

A105633 Row sums of triangle A105632.

Original entry on oeis.org

1, 2, 4, 9, 22, 57, 154, 429, 1223, 3550, 10455, 31160, 93802, 284789, 871008, 2681019, 8298933, 25817396, 80674902, 253106837, 796968056, 2517706037, 7977573203, 25347126630, 80738862085, 257778971504, 824798533933

Offset: 0

Paul D. Hanna, Apr 17 2005

nonn

Binomial transform of A007477. INVERT transform of A082582. First differences give A086581 and A025242 (offset 1). Is this sequence equal to A057580?
a(n) = the number of Dyck paths of semilength n+1 avoiding UUDU. a(n) = the number of Dyck paths of semilength n+1 avoiding UDUU = the number of binary trees without zigzag (i.e., with no node with a father, with a right son and with no left son). This sequence is the first column of the triangle A116424. E.g., a(2) = 4 because there exist four Dyck paths of semilength 3 that avoid UUDU: UDUDUD, UDUUDD, UUDDUD, UUUDDD, as well as four Dyck paths of semilength 3 that avoid UDUU: UDUDUD, UUDUDD, UUDDUD, UUUDDD. - I. Tasoulas (jtas(AT)unipi.gr), Feb 15 2006
The sequence beginning 1,1,2,4,9,... gives the diagonal sums of A130749, and has g.f. 1/(1-x-x^2/(1-x/(1-x-x^2/(1-x/(1-x-x^2/(1-... (continued fraction); and general term Sum_{k=0..floor(n/2)} Sum_{j=0..n-k} binomial(n-k,j)*A090181(j,k). Its Hankel transform is A099443(n+1). - Paul Barry, Jun 30 2009
The number of plain lambda terms presented by de Bruijn indices, see Bendkowski et al. - Kellen Myers, Jun 15 2015
a(n) = the number of Dyck paths of semilength n+1 with no pairs of
consecutive valleys at the same height. Sergi Elizalde, Feb 25 2021

			G.f.: A(x) = 1 + 2*x + 4*x^2 + 9*x^3 + 22*x^4 + 57*x^5 + 154*x^6 + 429*x^7 + ...
with A(x)^2 = 1 + 4*x + 12*x^2 + 34*x^3 + 96*x^4 + 274*x^5 + 793*x^6 + ...
where A(x) = 1 + x*(2-x)*A(x) + x^2*(1-x)*A(x)^2.
The logarithm of the g.f. begins:
log(A(x)) = (1 + (1-x))*x + (1 + 2^2*(1-x) + (1-x)^2)*x^2/2 +
(1 + 3^2*(1-x) + 3^2*(1-x)^2 + (1-x)^3)*x^3/3 +
(1 + 4^2*(1-x) + 6^2*(1-x)^2 + 4^2*(1-x)^3 + (1-x)^4)*x^4/4 +
(1 + 5^2*(1-x) + 10^2*(1-x)^2 + 10^2*(1-x)^3 + 5^2*(1-x)^4 + (1-x)^5)*x^5/5 + ...
Explicitly,
log(A(x)) = 2*x + 4*x^2/2 + 11*x^3/3 + 32*x^4/4 + 97*x^5/5 + 301*x^6/6 + 947*x^7/7 + 3008*x^8/8 + 9623*x^9/9 + 30959*x^10/10 + ...

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Andrei Asinowski, Axel Bacher, Cyril Banderier, and Bernhard Gittenberger, Analytic combinatorics of lattice paths with forbidden patterns, the vectorial kernel method, and generating functions for pushdown automata, Laboratoire d'Informatique de Paris Nord (LIPN 2019).
Marilena Barnabei, Flavio Bonetti, Niccolò Castronuovo, and Matteo Silimbani, Permutations avoiding a simsun pattern, The Electronic Journal of Combinatorics (2020) Vol. 27, Issue 3, P3.45. (See a_n in Theorem 4.)
Jean-Luc Baril, Daniela Colmenares, José L. Ramírez, Emmanuel D. Silva, Lina M. Simbaqueba, and Diana A. Toquica, Consecutive pattern-avoidance in Catalan words according to the last symbol, Univ. Bourgogne (France 2023).
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, 8.
Jean Luc Baril, Rigoberto Flórez, and José L. Ramirez, Generalized Narayana arrays, restricted Dyck paths, and related bijections, Univ. Bourgogne (France, 2025). See p. 10.
Paul Barry, Riordan Pseudo-Involutions, Continued Fractions and Somos 4 Sequences, arXiv:1807.05794 [math.CO], 2018. See p. 8.
Maciej Bendkowski, Quantitative aspects and generation of random lambda and combinatory logic terms, Ph.D. Thesis, Jagiellonian University, Kraków, 2017.
Maciej Bendkowski, Katarzyna Grygiel, Pierre Lescanne, and Marek Zaionc, Combinatorics of λ-terms: a natural approach, arXiv:1609.08106 [cs.LO], 2016.
Maciej Bendkowski, Katarzyna Grygiel, Pierre Lescanne, and Marek Zaionc, A Natural Counting of Lambda Terms, arXiv preprint arXiv:1506.02367 [cs.LO], 2015.
Maciej Bendkowski, K. Grygiel, and P. Tarau, Random generation of closed simply-typed lambda-terms: a synergy between logic programming and Boltzmann samplers, arXiv preprint arXiv:1612.07682 [cs.LO], 2016-2017.
David Callan, On Ascent, Repetition and Descent Sequences, arXiv:1911.02209 [math.CO], 2019. See p. 12.
Sergi Elizalde, Symmetric peaks and symmetric valleys in Dyck paths, arXiv:2008.05669 [math.CO], 2020. See p. 5.
Juan B. Gil and Michael D. Weiner, On pattern-avoiding Fishburn permutations, arXiv:1812.01682 [math.CO], 2018. See pp. 4, 6.
Katarzyna Grygiel and Pierre Lescanne, A natural counting of lambda terms, Preprint 2015.
Nancy S. S. Gu, Nelson Y. Li, and Toufik Mansour, 2-Binary trees: bijections and related issues, Discr. Math., 308 (2008), 1209-1221.
Toufik Mansour, Statistics on Dyck Paths, Journal of Integer Sequences, Vol. 9 (2006), Article 06.1.5.
A. Sapounakis et al., Ordered trees and the inorder transversal, Disc. Math., 306 (2006), 1732-1741.
Robin D. P. Zhou, Pattern avoidance in revised ascent sequences, arXiv:2505.05171 [math.CO], 2025. See pp. 4, 20.

Cf. A105632, A057580, A116424, A216604. See also A258973.

a := n -> add((-1)^i*hypergeom([(i+1)/2, i/2+1, i-n-1], [1, 2], -4), i=0..n+1):
seq(simplify(a(n)), n=0..26); # Peter Luschny, May 03 2018

CoefficientList[Series[(1 - x - Sqrt[(1 - x)^2 - 4 x^2/(1 - x)])/(2 x^2), {x, 0, 40}], x] (* Vincenzo Librandi, Mar 15 2014 *)

{a(n)=local(X=x+x*O(x^n)); polcoeff(2/(1-X)/(1-X+sqrt((1-X)^2-4*X^2/(1-X))),n,x)}

{a(n)=polcoeff(exp(sum(m=1,n+1,x^m/m*sum(k=0,m,binomial(m,k)^2*(1-x)^(m-k) + x*O(x^n)))),n)} \\ Paul D. Hanna, Sep 12 2012

G.f.: A(x) = (1-x - sqrt((1-x)^2 - 4*x^2/(1-x)))/(2*x^2).
a(n) = 2*a(n-1) + Sum_{i=1..n-2} a(i)*(a(n-1-i) - a(n-2-i)). a(n) = Sum_{i=0..floor(n/2)} (-1)^i * binomial(n+1-i,i) * binomial(2*(n+1)-3*i, n-2*i) /(n+1-i). - I. Tasoulas (jtas(AT)unipi.gr), Feb 15 2006
G.f.: (1/(1-x)^2)c(x^2/(1-x)^3), where c(x) is the g.f. of A000108. - Paul Barry, May 22 2009
1/(1-x-x/(1-x^2/(1-x-x/(1-x^2/(1-x-x/(1-x^2/(1-... (continued fraction). - Paul Barry, Jun 30 2009
a(n) = Sum_{k=0..floor(n/2)} Sum_{j=0..n-k} binomial(n-k,j)(0^(j+k)+(1/(j+0^j))*binomial(j,k)*binomial(j,k+1)). - Paul Barry, Jun 30 2009
G.f. satisfies: A(x) = (1 + x*A(x)) * (1 + x*(1-x)*A(x)). - Paul D. Hanna, Sep 12 2012
G.f.: exp( Sum_{n>=1} x^n/n * Sum_{k=0..n} binomial(n,k)^2 * (1-x)^k ). - Paul D. Hanna, Sep 12 2012
D-finite with recurrence: (n+2)*a(n) + (-4*n-3)*a(n-1) + (2*n+1)*a(n-2) + a(n-3) + (n-3)*a(n-4) = 0. - R. J. Mathar, Nov 26 2012
The recurrence is true, since by holonomic transformation, it can be computed formally using GFUN, associated with the equation: x^3 + x^2 - 2x + (x^3 + 3 x^2 -3x +1) A(x) + (x^5 + 2x^3 -4 x^2 + x) A'(x) = 0. - Pierre Lescanne, Jun 30 2015
G.f.: (1 - 1/(G(0)-x))/x^2 where G(k) =  1 + x/(1 + x/(x^2 - 1/G(k+1) )); (continued fraction). - Sergei N. Gladkovskii, Dec 16 2012
a(n) ~ 2^(n/3-1/6) * 3^(n+2) * (13+3*sqrt(33))^((n+1)/3) * sqrt(4*(2879 + 561*sqrt(33))^(1/3) + 8*(7822 + 1362*sqrt(33))^(1/3) - 91 - 21*sqrt(33)) / (((26+6*sqrt(33))^(2/3) - (26+6*sqrt(33))^(1/3) - 8)^(n+3/2) * (4*(26+6*sqrt(33))^(1/3) - (26+6*sqrt(33))^(2/3) + 8) * n^(3/2) * sqrt(Pi)). - Vaclav Kotesovec, Mar 13 2014
a(n) = Sum_{i=0..n+1} (-1)^i*hypergeom([(i+1)/2, i/2+1, i-n-1], [1, 2], -4). - Peter Luschny, May 03 2018

More terms from I. Tasoulas (jtas(AT)unipi.gr), Feb 15 2006

A175136 Triangle T(n,k) read by rows: number of LCO forests of size n with k leaves, 1 <= k <= n.

Original entry on oeis.org

1, 1, 1, 2, 2, 1, 4, 6, 3, 1, 8, 17, 12, 4, 1, 16, 46, 44, 20, 5, 1, 32, 120, 150, 90, 30, 6, 1, 64, 304, 482, 370, 160, 42, 7, 1, 128, 752, 1476, 1412, 770, 259, 56, 8, 1, 256, 1824, 4344, 5068, 3402, 1428, 392, 72, 9, 1, 512, 4352, 12368, 17285, 14000, 7168, 2436

Offset: 1

R. J. Mathar, Feb 21 2010

From Johannes W. Meijer, May 06 2011: (Start)
The Row1, Kn11, Kn12, Kn13, Kn21, Kn22, Kn23, Kn3, Kn4 and Ca1 triangle sums link A175136 with several sequences, see the crossrefs. For the definitions of these triangle sums see A180662.
It is remarkable that the coefficients of the right hand columns of A175136, and subsequently those of triangle A175136, can be generated with the aid of the row coefficients of A091894. For the fourth, fifth and sixth right hand columns see A162148, A190048 and A190049. The a(n) formulas of the right hand columns lead to an explicit formula for the T(n,k), see the formulas and the second Maple program. (End)
Triangle T(n,k), 1 <= k <= n, read by rows, given by (0,1,1,0,1,1,0,1,1,0,1,1,0,1,...) DELTA (1,0,0,1,0,0,1,0,0,1,0,0,1,0,...) where DELTA is the operator defined in A084938. - Philippe Deléham, Oct 29 2011.
T(n,k) is the number of noncrossing partitions of n containing k runs, where a block forms a run if it consists of an interval of integers. For example, T(4,2)=6 counts 1/234, 12/34, 123/4, 1/24/3, 13/2/4, 14/2/3. - David Callan, Oct 14 2012

			Triangle starts
    1;
    1,    1;
    2,    2,    1;
    4,    6,    3,    1;
    8,   17,   12,    4,    1;
   16,   46,   44,   20,    5,    1;
   32,  120,  150,   90,   30,    6,    1;
   64,  304,  482,  370,  160,   42,    7,    1;
  128,  752, 1476, 1412,  770,  259,   56,    8,    1;
Triangle (0,1,1,0,1,1,0,...) DELTA (1,0,0,1,0,0,1,...) begins:
  1;
  0,  1;
  0,  1,  1;
  0,  2,  2,  1;
  0,  4,  6,  3,  1;
  0,  8, 17, 12,  4,  1; ... - _Philippe Deléham_, Oct 29 2011

David Callan, A bijection on Dyck paths and its cycle structure, El. J. Combinat. 14 (2007) # R28.
David Callan, On Ascent, Repetition and Descent Sequences, arXiv:1911.02209 [math.CO], 2019.
David Callan and Emeric Deutsch, The Run Transform, arXiv:1112.3639 [math.CO], 2011.
K. Manes, A. Sapounakis, I. Tasoulas, and P. Tsikouras, Nonleft peaks in Dyck paths: a combinatorial approach, Discrete Math., 337 (2014), 97-105.

Triangle columns: A000012, A000027, A002378, A162148, A190048, A190049, A011782, A190050, A190051.
Triangle sums (see the comments): A000108 (Row1), A005043 (Related to Kn11, Kn12, Kn13 and Kn4), A007477 (Related to Kn21, Kn22, Kn23 and Kn3), A099251 (Kn4), A166300 (Ca1). - Johannes W. Meijer, May 06 2011
Cf. A000108 (row sums), A196182

lco := proc(siz,leav) (1-(1-4*x*(1-x)/(1-x*y))^(1/2))/2/x ; coeftayl(%,x=0,siz ) ; coeftayl(%,y=0,leav ) ; end proc: seq(seq(lco(n,k),k=1..n),n=1..9) ;
T := proc(n, k): add(A091894(n-k, k1)*binomial(n-k1-1, n-k), k1=0..floor((n-k)/2)) end: A091894 := proc(n, k): if n=0 and k=0 then 1 elif n=0 then 0 else 2^(n-2*k-1)* binomial(n-1, 2*k) * binomial(2*k, k)/(k+1) fi end: seq(seq(T(n, k), k=1..n), n=1..10); # Johannes W. Meijer, May 06 2011, revised Nov 23 2012

A091894[n_, k_] := 2^(n - 2*k - 1)*Binomial[n - 1, 2*k]*(Binomial[2*k, k]/(k + 1)); t[n_, k_] := Sum[A091894[n - k, k1]*Binomial [n - k1 - 1, n - k], {k1, 0, (n - k)/2}]; t[n_, n_] = 1; Table[t[n, k], {n, 1, 11}, {k, 1, n}] // Flatten(* Jean-François Alcover, Jun 13 2013, after Johannes W. Meijer *)

G.f.: (1-(1-4*x*(1-x)/(1-x*y))^(1/2))/(2*x).

T(n,k) = Sum_{k1=0..floor((n-k)/2)} A091894(n-k, k1)*binomial(n-k1-1, n-k), 1 <= k <= n. - Johannes W. Meijer, May 06 2011

Variable names changed by Johannes W. Meijer, May 06 2011

A307971 G.f. A(x) satisfies: A(x) = 1 + x + x^2 + x^3 + x^4*A(x)^2.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 3, 4, 5, 8, 13, 20, 29, 44, 70, 112, 175, 272, 430, 690, 1107, 1766, 2822, 4542, 7347, 11886, 19222, 31150, 50647, 82518, 134542, 219542, 358808, 587430, 962898, 1579686, 2593967, 4264292, 7017800, 11559548, 19055420, 31437318, 51908076, 85775954, 141841207

Offset: 0

Ilya Gutkovskiy, May 08 2019

nonn

Shifts 4 places left when convolved with itself.

			G.f.: A(x) = 1 + x + x^2 + x^3 + x^4 + 2*x^5 + 3*x^6 + 4*x^7 + 5*x^8 + 8*x^9 + 13*x^10 + ...

Seiichi Manyama, Table of n, a(n) for n = 0..4000

Cf. A000108, A007477, A307970, A307972.

a:= proc(n) option remember; `if`(n<4, 1,
      add(a(j)*a(n-4-j), j=0..n-4))
    end:
seq(a(n), n=0..50);  # Alois P. Heinz, May 08 2019

terms = 44; A[] = 0; Do[A[x] = 1 + x + x^2 + x^3 + x^4 A[x]^2 + O[x]^(terms + 1) // Normal, terms + 1]; CoefficientList[A[x], x]
a[n_] := a[n] = Sum[a[k] a[n - k - 4], {k, 0, n - 4}]; a[0] = a[1] = a[2] = a[3] = 1; Table[a[n], {n, 0, 44}]
CoefficientList[Series[(1 - Sqrt[1 - 4*x^4 - 4*x^5 - 4*x^6 - 4*x^7])/(2*x^4), {x, 0, 40}], x] (* Vaclav Kotesovec, Sep 27 2023 *)

G.f.: 1/(1 - x/(1 - x^4/(1 - x^4/(1 - x/(1 - x^4/(1 - x^4/(1 - x/(1 - x^4/(1 - x^4/(1 - ...)))))))))), a continued fraction.
Recurrence: a(n+4) = Sum_{k=0..n} a(k)*a(n-k).
G.f.: (1 - sqrt(1 - 4*x^4 - 4*x^5 - 4*x^6 - 4*x^7))/(2*x^4). - Vaclav Kotesovec, Sep 27 2023

A307972 G.f. A(x) satisfies: A(x) = 1 + x + x^2 + x^3 + x^4 + x^5*A(x)^2.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 2, 3, 4, 5, 6, 9, 14, 21, 30, 41, 58, 86, 130, 195, 286, 416, 612, 915, 1380, 2076, 3102, 4627, 6932, 10452, 15818, 23931, 36148, 54600, 82642, 125435, 190724, 290116, 441282, 671512, 1023052, 1560780, 2383578, 3642117, 5567202, 8514254, 13031192, 19960712

Offset: 0

Ilya Gutkovskiy, May 08 2019

nonn

Shifts 5 places left when convolved with itself.

			G.f.: A(x) = 1 + x + x^2 + x^3 + x^4 + x^5 + 2*x^6 + 3*x^7 + 4*x^8 + 5*x^9 + 6*x^10 + ...

Seiichi Manyama, Table of n, a(n) for n = 0..5000

Cf. A000108, A007477, A307970, A307971.

a:= proc(n) option remember; `if`(n<5, 1,
      add(a(j)*a(n-5-j), j=0..n-5))
    end:
seq(a(n), n=0..50);  # Alois P. Heinz, May 08 2019

terms = 47; A[] = 0; Do[A[x] = 1 + x + x^2 + x^3 + x^4 + x^5 A[x]^2 + O[x]^(terms + 1) // Normal, terms + 1]; CoefficientList[A[x], x]
a[n_] := a[n] = Sum[a[k] a[n - k - 5], {k, 0, n - 5}]; a[0] = a[1] = a[2] = a[3] = a[4] = 1; Table[a[n], {n, 0, 47}]

@CachedFunction
def a(n): # a = A307972
    if (n<5): return 1
    else: return sum(a(k)*a(n-k-5) for k in range(n-4))
[a(n) for n in range(51)] # G. C. Greubel, Nov 26 2022

G.f.: 1/(1 - x/(1 - x^5/(1 - x^5/(1 - x/(1 - x^5/(1 - x^5/(1 - x/(1 - x^5/(1 - x^5/(1 - ...)))))))))), a continued fraction.

Recurrence: a(n+5) = Sum_{k=0..n} a(k)*a(n-k).

A025242 Generalized Catalan numbers A(x)^2 -(1+x)^2A(x) +x(2+x+x^2) =0.

Original entry on oeis.org

2, 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: 1

Clark Kimberling, Olivier Gérard

Number of Dyck paths of semilength n-1 with no UUDD (n>1). Example: a(4)=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
a(n) is the number of Dyck (n-2)-paths with no DDUU (n>2). Example: a(6)=13 counts all 14 Dyck 4-paths except UUDDUUDD which contains a DDUU. There is a simple bijective proof: given a Dyck path that avoids DDUU, for every occurrence of UUDD except the first, the ascent containing this UU must be immediately preceded by a UD (else a DDUU would be present). Transfer the latter UD to the middle of the DD in the UUDD. Then insert a new UD in the middle of the first DD if any; if not, the path is a sawtooth UDUD...UD, in which case insert a UD at the end. This is a bijection from DDUU-avoiding Dyck n-paths to UUDD-avoiding Dyck (n+1)-paths. - David Callan, Sep 25 2006
For n>1, a(n) is the number of cyclic permutations of [n-1] that avoid the vincular pattern 13-4-2, i.e., the pattern 1342 where the 1 and 3 must be adjacent. By the trivial Wilf equivalence, the same applies for 24-3-1, 31-2-4, and 42-1-3. - Rupert Li, Jul 27 2021

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Sela Fried, Even-up words and their variants, arXiv:2505.14196 [math.CO], 2025. See p. 10.
Petr Gregor, Torsten Mütze, and Namrata, Combinatorial generation via permutation languages. VI. Binary trees, arXiv:2306.08420 [cs.DM], 2023.
Petr Gregor, Torsten Mütze, and Namrata, Pattern-Avoiding Binary Trees-Generation, Counting, and Bijections, Leibniz Int'l Proc. Informatics (LIPIcs), 34th Int'l Symp. Algor. Comp. (ISAAC 2023). See p. 33.13.
Nancy S. S. Gu, Nelson Y. Li, and Toufik Mansour, 2-Binary trees: bijections and related issues, Discr. Math., 308 (2008), 1209-1221.
Jia Huang and Erkko Lehtonen, Associative-commutative spectra for some varieties of groupoids, arXiv:2401.15786 [math.CO], 2024. See p. 18.
Vít Jelínek, Toufik Mansour and Mark Shattuck, On multiple pattern avoiding set partitions, Adv. Appl. Math. 50 (2) (2013) 292-326, Theorem 4.1, without the leading 2.
Yvan Le Borgne, Counting Upper Interactions in Dyck Paths, Séminaire Lotharingien de Combinatoire, Vol. 54, B54f (2006), 16 pp.
Rupert Li, Vincular Pattern Avoidance on Cyclic Permutations, arXiv:2107.12353 [math.CO], 2021.
Toufik Mansour, Restricted 1-3-2 permutations and generalized patterns, arXiv:math/0110039 [math.CO], 2001.
Toufik Mansour, Restricted 1-3-2 permutations and generalized patterns, Annals of Combin., 6 (2002), 65-76. (Example 2.10.)
Toufik Mansour and Mark Shattuck, Restricted partitions and generalized Catalan numbers, PU. M. A., Vol. (2011), No. 2, pp. 239-251. - From _N. J. A. Sloane_, Oct 13 2012
Lara Pudwell, Pattern-avoiding ascent sequences, Slides from a talk, 2015.
Lara Pudwell and Andrew Baxter, Ascent sequences avoiding pairs of patterns, Slides, Permutation Patterns 2014, East Tennessee State University Jul 07 2014.
A. Sapounakis, I. Tasoulas and P. Tsikouras, Counting strings in Dyck paths, Discrete Math., 307 (2007), 2909-2924.

Cf. A000108, A001006, A006318, A004148, A007477, A082582, A086581.

a[ 0 ]=1; a[ n_Integer ] := a[ n ]=a[ n-1 ]+Sum[ a[ k ]*a[ n-1-k ], {k, 2, n-1} ];

a(n)=polcoeff((1+2*x+x^2-sqrt(1-4*x+2*x^2+x^4+x*O(x^n)))/2,n)

a(n) = a(1)*a(n-1) + a(2)*a(n-2) + ... + a(n-3)*a(3) for n >= 4.
G.f.: (1+2*x+x^2-sqrt(1-4*x+2*x^2+x^4))/2. - Michael Somos, Jun 08 2000
Conjecture: n*(n+1)*a(n) +(n^2+n+2)*a(n-1) +2*(-9*n^2+15*n+17)*a(n-2) +2*(5*n+4)*(n-4)*a(n-3) +(n+1)*(n-6)*a(n-4) +(5*n+4)*(n-7)*a(n-5)=0. - R. J. Mathar, Jan 12 2013
G.f.: 2 + 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

A115178 Expansion of c(x^2+x^3), c(x) the g.f. of A000108.

Original entry on oeis.org

1, 0, 1, 1, 2, 4, 7, 15, 29, 61, 126, 266, 566, 1212, 2619, 5685, 12419, 27247, 60049, 132847, 294931, 656877, 1467258, 3286218, 7378240, 16603458, 37441990, 84599854, 191501532, 434224404, 986161959, 2243009869, 5108859821

Offset: 0

Paul Barry, Mar 14 2006

Diagonal sums of number triangle A117434.
a(n) = number of Motzkin n-paths (A001006) in which every flatstep (F) is followed by a downstep (D). For example, a(5)=4 counts UDUFD, UFDUD, UUDFD, UUFDD. - David Callan, Jun 07 2006
a(n) = number of lattice paths in the first quadrant from (0,0) to (n,0) using only steps U1=(1,1), U2=(2,1) and D=(1,-1). E.g. a(6)=7 because we have U1DU1DU1D, U1U1U1DDD, U1U1DU1DD, U1DU1U1DD, U1U1DDU1D, U2DU2D and U2U2DD. - José Luis Ramírez Ramírez, May 27 2013

			1 + x^2 + x^3 + 2*x^4 + 4*x^5 + 7*x^6 + 15*x^7 + 29*x^8 + 61*x^9 + ...

G. C. Greubel, Table of n, a(n) for n = 0..1000

Cf. A007477.

Table[Sum[Binomial[k, n - 2*k]*CatalanNumber[k], {k, 0, Floor[n/2]}], {n, 0, 50}] (* G. C. Greubel, Feb 03 2017 *)

{a(n) = local(A); A = O(x^0); for( k=0, n\5, A = 1 / (1 - x^2 / (1 - x / (1 - x^2 * A)))); polcoeff( A, n)} /* Michael Somos, May 12 2012 */

a(n) = Sum_{k=0..floor(n/2)} C(k)*C(k,n-2k).
D-finite with recurrence (n+2)*a(n) +(n+2)*a(n-1) +4*(1-n)*a(n-2) +2*(7-4*n)*a(n-3) +2*(5-2*n)*a(n-4)=0. - R. J. Mathar, Nov 15 2011
G.f. A(x) satisfies A(x) = 1 / (1 - x^2 / (1 - x / (1 - x^2 * A(x)))). - Michael Somos, May 12 2012
G.f.: (1-sqrt(1-4*z^2*(1+z)))/(2*z^2*(1+z)). - José Luis Ramírez Ramírez, May 27 2013
a(n) ~ sqrt(3 - 1/9*(-2 + (19-3*sqrt(33))^(1/3) + (19+3*sqrt(33))^(1/3))^2) * (((-2 + (19-3*sqrt(33))^(1/3) + (19+3*sqrt(33))^(1/3)) * (4 + (19-3*sqrt(33))^(1/3) + (19+3*sqrt(33))^(1/3)))/9)^n /(n^(3/2)*sqrt(Pi)). - Vaclav Kotesovec, Sep 16 2013

A307970 G.f. A(x) satisfies: A(x) = 1 + x + x^2 + x^3*A(x)^2.

Original entry on oeis.org

1, 1, 1, 1, 2, 3, 4, 7, 12, 19, 32, 56, 96, 165, 290, 512, 902, 1601, 2862, 5124, 9198, 16585, 29990, 54336, 98702, 179742, 327942, 599432, 1097756, 2013737, 3699596, 6806866, 12541518, 23137270, 42736850, 79031394, 146309968, 271142469, 502978944, 933921458, 1735634266

Offset: 0

Ilya Gutkovskiy, May 08 2019

nonn

Shifts 3 places left when convolved with itself.

			G.f.: A(x) = 1 + x + x^2 + x^3 + 2*x^4 + 3*x^5 + 4*x^6 + 7*x^7 + 12*x^8 + 19*x^9 + 32*x^10 + ...

Seiichi Manyama, Table of n, a(n) for n = 0..3000

Cf. A000108, A007477, A307971, A307972.

a:= proc(n) option remember; `if`(n<3, 1,
      add(a(j)*a(n-3-j), j=0..n-3))
    end:
seq(a(n), n=0..50);  # Alois P. Heinz, May 08 2019

terms = 40; A[] = 0; Do[A[x] = 1 + x + x^2 + x^3 A[x]^2 + O[x]^(terms + 1) // Normal, terms + 1]; CoefficientList[A[x], x]
a[n_] := a[n] = Sum[a[k] a[n - k - 3], {k, 0, n - 3}]; a[0] = a[1] = a[2] = 1; Table[a[n], {n, 0, 40}]

G.f.: 1/(1 - x/(1 - x^3/(1 - x^3/(1 - x/(1 - x^3/(1 - x^3/(1 - x/(1 - x^3/(1 - x^3/(1 - ...)))))))))), a continued fraction.
Recurrence: a(n+3) = Sum_{k=0..n} a(k)*a(n-k).
a(n) ~ sqrt(3 + 4*r^4 + 8*r^5) / (4*sqrt(Pi)*n^(3/2)*r^(n+3)), where r = 0.51899425841331458784223152875297289010563957455264491744143... is the root of the equation 1 + r + r^2 = 1/(4*r^3). - Vaclav Kotesovec, Jul 03 2021

cp's OEIS Frontend

A213705 a(n)=n if n <= 3, otherwise a(n) = A007477(n-1) + A007477(n).

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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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Maxima

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A105633 Row sums of triangle A105632.

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A175136 Triangle T(n,k) read by rows: number of LCO forests of size n with k leaves, 1 <= k <= n.

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A307971 G.f. A(x) satisfies: A(x) = 1 + x + x^2 + x^3 + x^4*A(x)^2.

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A307972 G.f. A(x) satisfies: A(x) = 1 + x + x^2 + x^3 + x^4 + x^5*A(x)^2.

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

A025242 Generalized Catalan numbers A(x)^2 -(1+x)^2A(x) +x(2+x+x^2) =0.