Wenjin Woan - cp's OEIS frontend

User: Wenjin Woan

Wenjin Woan has authored 8 sequences.

A131178 Non-plane increasing unary binary (0-1-2) trees where the nodes of outdegree 1 come in 2 colors.

1, 2, 5, 16, 64, 308, 1730, 11104, 80176, 643232, 5676560, 54650176, 569980384, 6401959328, 77042282000, 988949446144, 13488013248256, 194780492544512, 2969094574403840, 47640794742439936, 802644553810683904, 14166772337295285248, 261410917571703825920

Offset: 1

Wenjin Woan, Oct 31 2007

A labeled tree of size n is a rooted tree on n nodes that are labeled by distinct integers from the set {1,...,n}. An increasing tree is a labeled tree such that the sequence of labels along any branch starting at the root is increasing. Thus the root of an increasing tree will be labeled 1. In unary binary trees (sometimes called 0-1-2 trees) the outdegree of a node is either 0, 1 or 2. Here we are counting non-plane (where the subtrees stemming from a node are not ordered between themselves) increasing unary binary trees where the nodes of outdegree 1 come in two colors. An example is given below. - Peter Bala, Sep 01 2011
The number of plane increasing 0-1-2 trees on n nodes, where the nodes of outdegree 1 come in two colors, is equal to n!. Other examples of sequences counting increasing trees include A000111, A000670, A008544, A008545, A029768 and A080635. - Peter Bala, Sep 01 2011
Number of plane increasing 0-1-2 trees, where the nodes of outdegree 1 come in 2 colors, avoiding pattern T213. See A278679 for more definitions and examples. - Sergey Kirgizov, Dec 24 2016

			G.f. = x + 2*x^2 + 5*x^3 + 16*x^4 + 64*x^5 + 308*x^6 + 1730*x^7 + 11104*x^8 + ...
a(3) = 5: Denoting the two types of node of outdegree 1 by the letters a or b, the 5 possible trees are
.
.  1a    1b    1a    1b      1
.  |     |     |     |      / \
.  2a    2b    2b    2a    2   3
.  |     |     |     |
.  3     3     3     3
- _Peter Bala_, Sep 01 2011

Vincenzo Librandi, Table of n, a(n) for n = 1..100
Jean-Luc Baril, Sergey Kirgizov, Vincent Vajnovszki, Patterns in treeshelves, arXiv:1611.07793 [cs.DM], 2016.
F. Bergeron, Ph. Flajolet and B. Salvy, Varieties of Increasing Trees, Lecture Notes in Computer Science vol. 581, ed. J.-C. Raoult, Springer 1992, pp. 24-48.
Lapo Cioni, Luca Ferrari, and Corentin Henriet. A direct bijection between two-stack sortable permutations and fighting fish, Euro. Conf. Comb., Graph Theory Appl. (2023) No. 12, 283-289.
D. Dominici, Nested derivatives: A simple method for computing series expansions of inverse functions. arXiv:math/0501052v2 [math.CA], 2005.

Cf. A000111, A000670, A008544, A008545, A029768, A080635, A278679

E:=  (2*(exp(sqrt(2)*x)-1)) / ((2+sqrt(2))-(2-sqrt(2))*exp(sqrt(2)*x)):
S:= map(simplify,series(E,x,101)):
seq(coeff(S,x,j)*j!, j=1..100); # Robert Israel, Nov 23 2016

max = 25; f[x_] := (2*(Exp[Sqrt[2]*x] - 1))/((2 + Sqrt[2]) - (2 - Sqrt[2])*Exp[Sqrt[2]*x]); Drop[ Simplify[ CoefficientList[ Series[f[x], {x, 0, max}], x]*Range[0, max]!], 1] (* Jean-François Alcover, Oct 05 2011 *)

x='x+O('x^66); /* that many terms */
default(realprecision,1000); /* working with floats here */
egf=(2*(exp(sqrt(2)*x)-1)) / ((2+sqrt(2))-(2-sqrt(2))*exp(sqrt(2)*x));
round(Vec(serlaplace(egf))) /* show terms */
/* Joerg Arndt, Sep 01 2011 */

/* the following program should be preferred. */
Vec( serlaplace( serreverse( intformal( 1/(1+2*x+1/2*x^2) + O(x^66) ) ) ) )
\\ Joerg Arndt, Mar 01 2014

{a(n) = if( n<1, 0, n! * polcoeff( 2 / (-2 + quadgen(8) * (-1 + 2 / (1 - exp(-quadgen(8) * x + x * O(x^n))))), n))};

E.g.f.: A(x) = (2*(exp(sqrt(2)*x)-1)) / ((2+sqrt(2))-(2-sqrt(2))*exp(sqrt(2)*x)) = x+2*x^2/2!+5*x^3/3!+16*x^4/4!+64*x^5/5!+....
From Peter Bala, Sep 01 2011: (Start)
The generating function A(x) satisfies the autonomous differential equation A' = 1+2*A+1/2*A^2 with A(0) = 0. It follows that the inverse function A(x)^-1 may be expressed as an integral A(x)^-1 = int {t = 0..x} 1/(1+2*t+1/2*t^2).
Applying [Dominici, Theorem 4.1] to invert the integral gives the following method for calculating the terms of the sequence: let f(x) = 1+2*x+1/2*x^2. Let D be the operator f(x)*d/dx. Then a(n) = D^n(f(x)) evaluated at x = 0. Compare with A000111(n+1) = D^n(1+x+x^2/2!) evaluated at x = 0.
(End)
G.f.: 1/Q(0), where Q(k) = 1 - 2*x*(2*k+1) - m*x^2*(k+1)*(2*k+1)/( 1 - 2*x*(2*k+2) - m*x^2*(k+1)*(2*k+3)/Q(k+1) ) and m=1; (continued fraction). - Sergei N. Gladkovskii, Sep 24 2013
G.f.: 1/Q(0), where Q(k) = 1 - 2*x*(k+1) - 1/2*x^2*(k+1)*(k+2)/Q(k+1); (continued fraction). - Sergei N. Gladkovskii, Oct 02 2013
a(n) ~ n! * 2^((n+3)/2) / log(3+2*sqrt(2))^(n+1). - Vaclav Kotesovec, Oct 08 2013
G.f.: conjecture: T(0)/(1-2*x) -1, where T(k) = 1 - x^2*(k+1)*(k+2)/(x^2*(k+1)*(k+2) - 2*(1-2*x*(k+1))*(1-2*x*(k+2))/Q(k+1) ); (continued fraction). - Sergei N. Gladkovskii, Nov 19 2013
E.g.f.: x/(T(0)-x), where T(k) = 4*k + 1 + x^2/(8*k+6 + x^2/T(k+1) ); (continued fraction). - Sergei N. Gladkovskii, Nov 30 2013

Terms >= 80176 from Peter Bala, Sep 01 2011

Changed offset to 1 to agree with name and example. - Michael Somos, Nov 23 2016

A131638 Increasing binary trees having exactly two vertices with outdegree 1.

Original entry on oeis.org

1, 11, 180, 4288, 141584, 6213288, 350400832, 24718075136, 2133652515072, 221311262045440, 27166907582280704, 3895974311462313984, 645512064907811491840, 122381396964887716078592, 26325690425815766552887296, 6377608610246241663568248832

Offset: 1

Wenjin Woan, Oct 03 2007

nonn

M. P. Develin and S. P. Sullivant, Markov Bases of Binary Graph Models, Annals of Combinatorics 7 (2003) 441-466.
Christiane Poupard, Deux propriétés des arbres binaires ordonnés stricts, Europ. J. Combin., vol. 10, 1989, pp. 369-374.

Cf. A002105, A094503.

Table[n!*SeriesCoefficient[1/2*(-((x*Sec[x/Sqrt[2]]^2 *Tan[x/Sqrt[2]]) /Sqrt[2]) + 3*Sec[x/Sqrt[2]]^2 *Tan[x/Sqrt[2]]^2), {x, 0, n}], {n, 2, 40, 2}] (* Vaclav Kotesovec after Michel Marcus, Sep 25 2013 *)

lista(m) = { default(realprecision, 30); x = y + O(y^m); egf = (3*tan(x/sqrt(2))^2/cos(x/sqrt(2))^2-x*tan(x/sqrt(2))/(sqrt(2)*cos(x/sqrt(2))^2))/2; forstep (n=2, m, 2, print1(round(n!*polcoeff(egf, n, y)), ", "));}  \\ Michel Marcus, Mar 03 2013

E.g.f.: (3*sec(x/sqrt(2))^2*tan(x/sqrt(2))^2-x*sec(x/sqrt(2))^2*tan(x/sqrt(2))/(sqrt(2)))/2. - Michel Marcus, Mar 03 2013
a(n) ~ (2*n)! * 2^(n+6)*n^3/Pi^(2*n+4). - Vaclav Kotesovec, Sep 25 2013
From Klaus K Haverkamp, Jul 02 2023: (Start)
a(n) = (A002105(n+2) - (n+1)*A002105(n+1))/2.
a(n) = A094503(2n+1,n). (End)

More terms from Michel Marcus, Mar 03 2013

A125307 Number of increasing trees with branches of height 1.

Original entry on oeis.org

1, 1, 4, 15, 80, 480, 3444, 27790, 253504, 2556792, 28382880, 343071168, 4490999424, 63253633872, 954133373088, 15343385194800, 262060291958784, 4737396899952384, 90370907329842432, 1814141041750834560, 38229440785429201920, 843786230514306621696

Offset: 1

Wenjin Woan, Jan 17 2007

nonn

R. P. Stanley, Enumerative Combinatorics, Vol. 1, Cambridge University Press, 1997. Proposition 1.3.16, p. 25.

G. C. Greubel, Table of n, a(n) for n = 1..448
Vladimir Kruchinin, D. V. Kruchinin, Composita and their properties, arXiv:1103.2582 [math.CO], 2011-2013.

Range[0, 21]!CoefficientList[ Series[(x - 1 + Log[1 - x])/((1 - x)^2(Log[1 - x] - 1)), {x, 0, 21}], x] (* Robert G. Wilson v, Jan 26 2007 *)

a(n):=n!*(sum((-1)^(m)*(n-m+1)/(m-1)!*sum(k!*stirling1(m-1,k), k,1,m-1), m,2,n)+1); /* Vladimir Kruchinin, Sep 09 2010 */

x='x+O('x^30); Vec(serlaplace( (x-1+log(1-x))/((x-1)^2*(log(1-x) -1)))) \\ G. C. Greubel, Sep 05 2018

E.g.f.: (x-1+log(1-x)) / ( (x-1)^2 (log(1-x)-1) ).
a(n) = n!*(sum((-1)^(m)*(n-m+1)/(m-1)!*sum(k!*Stirling1(m-1,k),k,1,m-1),m,2,n)+1). - Vladimir Kruchinin, Sep 09 2010
a(n) ~ n!*n*(1 - 1/log(n) + gamma/log(n)^2), where gamma is Euler-Mascheroni constant (A001620). - Vaclav Kotesovec, Sep 25 2013

More terms from N. J. A. Sloane, Jan 26 2007

A125062 Number of increasing trees with hills of height 1.

Original entry on oeis.org

1, 1, 4, 15, 68, 370, 2364, 17388, 144864, 1349136, 13894560, 156831840, 1925527680, 25550778240, 364416917760, 5559659078400, 90349397913600, 1558170228787200, 28423674336153600, 546807873520742400, 11064204944529408000, 234902850943703040000, 5221386564941352960000

Offset: 0

Wenjin Woan, Jan 09 2007

nonn

If we discard the first 1 and set a(0)=1,a(1)=4, then a(n) = (n+1)!(H(n)+1), where H(n) = Sum_{k=1..n} 1/k. - Gary Detlefs, Jul 21 2010

R. P. Stanley, Enumerative Combinatorics, Vol. 1, Cambridge University Press, 1997, p25.

Vincenzo Librandi, Table of n, a(n) for n = 0..200

Cf. A001620.

a := n -> ifelse(n = 0, 1, (n - 1)! * (n*(harmonic(n) + 1) - 1)):
seq(a(n), n = 0..22);  # Peter Luschny, Apr 09 2024

With[{nn=20},CoefficientList[Series[(1+x Log[1/(1-x)])/(1-x),{x,0,nn}], x]Range[0,nn]!] (* Harvey P. Dale, Mar 14 2012 *)
a[0]=1;a[n_]:=(n-1)*(n-1)!+Abs[StirlingS1[n+1,2]];Flatten[Table[a[n],{n,0,19}]] (* Detlef Meya, Apr 09 2024 *)

x='x+O('x^30); Vec(serlaplace((1+x*log(1/(1-x)))/(1-x))) \\ G. C. Greubel, Aug 31 2018

E.g.f.: (1+x*log(1/(1-x)))/(1-x).
a(n) = 2*(n-1)*a(n-1) - (n^2-4*n+5)*a(n-2) - (n-3)*(n-2)*a(n-3). - Vaclav Kotesovec, Nov 19 2012
a(n) ~ n!*(log(n) + gamma + 1 + O(log(n)/n)), where gamma is Euler-Mascheroni constant (A001620). - Vaclav Kotesovec, Nov 19 2012
a(0) = 1; For n > 0; a(n) = (n - 1)*(n - 1)! + abs(Stirling1(n + 1, 2)). - Detlef Meya, Apr 09 2024

Edited by the Associate Editors of the OEIS, Oct 05 2009

A079280 Number of log-concave paths of length n starting from the origin (0,0) with steps from {N=(0,1), E=(1,0) and S=(0,-1)} that stay in the second octant and never touch the line y=x except possibly at the beginning or the end.

Original entry on oeis.org

1, 2, 2, 5, 7, 17, 26, 62, 99, 233, 382, 890, 1486, 3434, 5812, 13340, 22819, 52073, 89846, 204002, 354522, 801422, 1401292, 3155300, 5546382, 12444842, 21977516, 49155332, 87167164, 194392628, 345994216, 769547192, 1374282019, 3049104233

Offset: 1

Wenjin Woan, Feb 07 2003

nonn

The first S step can only come after an E step.

G.f.: (1-x)/(2-4*x) + (1+3*x)/(2*sqrt(1-4*x^2));
a(2n) = (1/2)*(4^(n-1) + 3*binomial(2n-2, n-1));
a(2n+1) = (1/2)*(2*4^(n-1) + binomial(2n, n)).

More terms from Max Alekseyev, Jul 04 2009

A059435 Number of lattice paths in plane starting at (0,0) and ending at (n,n) with steps from {(i,j): i+j > 0, i, j >= 0} that never go below the line y = x.

Original entry on oeis.org

1, 2, 12, 88, 720, 6304, 57792, 547712, 5323008, 52761088, 531311616, 5420488704, 55905767424, 581954543616, 6106210615296, 64513688174592, 685741070942208, 7328106153115648, 78684992821788672, 848487859401261056

Offset: 0

Wenjin Woan, Feb 01 2001

Series reversion of x(1-4x)/(1-2x). - Paul Barry, May 19 2005

The Hankel transform of this sequence is 8^C(n+1,2) = [1, 8, 512, 262144, ...]. - Philippe Deléham, Nov 08 2007

W.-J. Woan, A bijective proof by induction that the n-th term of this sequence is 2^(n-1) times of the n-th term of the big Schroeder number, 2001 (unpublished).

Vincenzo Librandi, Table of n, a(n) for n = 0..200
David Callan, A uniformly distributed statistic on a class of lattice paths, Electronic J. Combinatorics, Vol. 11(1), R82, 2004.
Z. Chen and H. Pan, Identities involving weighted Catalan-Schroder and Motzkin Paths, arXiv:1608.02448 (2016); see Eq. (1.13) with a=2 and b=4.
Ira M. Gessel, A factorization for formal Laurent series and lattice path enumeration, J. Combin. Theory Ser. A 28 (1980), 321-337.
Elina Robeva and Melinda Sun, Bimonotone Subdivisions of Point Configurations in the Plane, arXiv:2007.00877 [math.CO], 2020. See B(2,n) column in Table 3 (p. 10).
Robert A. Sulanke, Counting lattice paths by Narayana polynomials, Electronic J. Combinatorics 7 (2000), R40.

Cf. A001003, A006318, A054726.

gf := (1+2*x-sqrt(4*x^2-12*x+1))/(8*x): s := series(gf, x, 100): for i from 0 to 50 do printf(`%d,`,coeff(s,x,i)) od:

Table[SeriesCoefficient[(1+2*x-Sqrt[4*x^2-12*x+1])/(8*x),{x,0,n}],{n,0,20}] (* Vaclav Kotesovec, Oct 11 2012 *)

x='x+O('x^66); Vec((1+2*x-sqrt(4*x^2-12*x+1))/(8*x)) \\ Joerg Arndt, May 06 2013

a(n) = 2^n*A001003(n).
G.f.: (1 + 2*x - sqrt(4*x^2 - 12*x + 1))/(8*x).
From Paul Barry, May 19 2005: (Start)
a(n) = (1/(n + 1)) * Sum_{k=0..n} C(n+1, k) * C(2*n-k, n)(-1)^k * 4^(n-k) * 2^k;
a(n) = Sum_{k=0..n} (1/n) * C(n, k) * C(n, k+1) * 4^k * 2^(n-k);
a(n) = Sum_{k=1..n} N(n, k)*2^(n+k-1), for n >= 1, where N(n, k) are the Narayana numbers (A001263). [Corrected by Alejandro H. Morales, May 14 2015]
(End)
Recurrence: (n+1)*a(n) = 6*(2*n-1)*a(n-1) - 4*(n-2)*a(n-2). - Vaclav Kotesovec, Oct 11 2012
a(n) ~ sqrt(4+3*sqrt(2))*(6+4*sqrt(2))^n/(4*sqrt(Pi)*n^(3/2)). - Vaclav Kotesovec, Oct 11 2012

A059231 Number of different lattice paths running from (0,0) to (n,0) using steps from S = {(k,k) or (k,-k): k positive integer} that never go below the x-axis.

Original entry on oeis.org

1, 1, 5, 29, 185, 1257, 8925, 65445, 491825, 3768209, 29324405, 231153133, 1841801065, 14810069497, 120029657805, 979470140661, 8040831465825, 66361595715105, 550284185213925, 4582462506008253, 38306388126997785, 321327658068506121, 2703925940081270205

Offset: 0

Wenjin Woan, Jan 20 2001

If y = x*A(x) then 4*y^2 - (1 + 3*x)*y + x = 0 and x = y*(1 - 4*y) / (1 - 3*y). - Michael Somos, Sep 28 2003
a(n) = A059450(n, n). - Michael Somos, Mar 06 2004
The Hankel transform of this sequence is 4^binomial(n+1,2). - Philippe Deléham, Oct 29 2007
a(n) is the number of Schroder paths of semilength n in which there are no (2,0)-steps at level 0 and at a higher level they come in 3 colors. Example: a(2)=5 because we have UDUD, UUDD, UBD, UGD, and URD, where U=(1,1), D=(1,-1), while B, G, and R are, respectively, blue, green, and red (2,0)-steps. - Emeric Deutsch, May 02 2011
Shifts left when INVERT transform applied four times. - Benedict W. J. Irwin, Feb 02 2016

			a(3) = 29 since the top row of Q^2 = (5, 8, 16, 0, 0, 0, ...), and 5 + 8 + 16 = 29.

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Paul Barry, On Integer-Sequence-Based Constructions of Generalized Pascal Triangles, Journal of Integer Sequences, Vol. 9 (2006), Article 06.2.4.
Paul Barry and Aoife Hennessy, A Note on Narayana Triangles and Related Polynomials, Riordan Arrays, and MIMO Capacity Calculations, J. Int. Seq. 14 (2011), Article 11.3.8.
Paul Barry and Aoife Hennessy, Generalized Narayana Polynomials, Riordan Arrays, and Lattice Paths, Journal of Integer Sequences, Vol. 15 (2012), Article 12.4.8.
Zhi Chen and Hao Pan, Identities involving weighted Catalan, Schroder and Motzkin paths, arXiv:1608.02448 [math.CO], 2016. See eq. (1.13), a=1, b=4.
Curtis Coker, Enumerating a class of lattice paths, Discrete Math., 271 (2003), 13-28 (the sequence d_n).
Curtis Coker, A family of eigensequences, Discrete Math. 282 (2004), 249-250.
Aoife Hennessy, A Study of Riordan Arrays with Applications to Continued Fractions, Orthogonal Polynomials and Lattice Paths, Ph. D. Thesis, Waterford Institute of Technology, Oct. 2011.
Joseph P. S. Kung and Anna de Mier, Catalan lattice paths with rook, bishop and spider steps, J. Comb. Theor., Series A (2013) Vol. 120, Issue 2, 379-389.
Gregory J. Morrow, Laws relating runs and steps in gambler’s ruin, Stochastic Proc. Appl. (2024) Vol. 125, Issue 5, 2010-2025.
Gregory Morrow, Some probability distributions and integer sequences related to rook paths, Univ. Colorado Springs (2024). See pp. 1, 4, 15, 22. DOI
Michael Z. Spivey and Laura L. Steil, The k-Binomial Transforms and the Hankel Transform, J. Integ. Seqs. Vol. 9 (2006), #06.1.1.
Paveł Szabłowski, Beta distributions whose moment sequences are related to integer sequences listed in the OEIS, Contrib. Disc. Math. (2024) Vol. 19, No. 4, 85-109. See p. 98.
Wen-jin Woan, Diagonal lattice paths, Congr. Numer. 151 (2001) 173-178.

Cf. A000108, A001003, A007564, A127846.
Row sums of A086873.
Column k=2 of A227578. - Alois P. Heinz, Jul 17 2013

gf := (1+3*x-sqrt(9*x^2-10*x+1))/(8*x): s := series(gf, x, 100): for i from 0 to 50 do printf(`%d,`,coeff(s, x, i)) od:
A059231_list := proc(n) local j, a, w; a := array(0..n); a[0] := 1;
for w from 1 to n do a[w] := a[w-1]+4*add(a[j]*a[w-j-1],j=1..w-1) od;
convert(a, list) end: A059231_list(20); # Peter Luschny, May 19 2011

Join[{1},Table[-I 3^n/2LegendreP[n,-1,5/3],{n,40}]] (* Harvey P. Dale, Jun 09 2011 *)
Table[Hypergeometric2F1[-n, 1 - n, 2, 4], {n, 0, 22}] (* Arkadiusz Wesolowski, Aug 13 2012 *)

{a(n) = if( n<0, 0, polcoeff( (1 + 3*x - sqrt(1 - 10*x + 9*x^2 + x^2 * O(x^n))) / (8*x), n))}; /* Michael Somos, Sep 28 2003 */

{a(n) = if( n<0, 0, n++; polcoeff( serreverse( x * (1 - 4*x) / (1 - 3*x) + x * O(x^n)), n))}; /* Michael Somos, Sep 28 2003 */

# Algorithm of L. Seidel (1877)
def A059231_list(n) :
    D = [0]*(n+2); D[1] = 1
    R = []; b = False; h = 1
    for i in range(2*n) :
        if b :
            for k in range(1, h, 1) : D[k] += 2*D[k+1]
        else :
            for k in range(h, 0, -1) : D[k] += 2*D[k-1]
            h += 1
        b = not b
        if b : R.append(D[1])
    return R
A059231_list(23)  # Peter Luschny, Oct 19 2012

a(n) = Sum_{k=0..n} 4^k*N(n, k) where N(n, k) = (1/n)*binomial(n, k)*binomial(n, k+1) are the Narayana numbers (A001263). - Benoit Cloitre, May 10 2003
a(n) = 3^n/2*LegendreP(n, -1, 5/3). - Vladeta Jovovic, Sep 17 2003
G.f.: (1 + 3*x - sqrt(1 - 10*x + 9*x^2)) / (8*x) = 2 / (1 + 3*x + sqrt(1 - 10*x + 9*x^2)). - Michael Somos, Sep 28 2003
a(n) = Sum_{k=0..n} A088617(n, k)*4^k*(-3)^(n-k). - Philippe Deléham, Jan 21 2004
With offset 1: a(1)=1, a(n) = -3*a(n-1) + 4*Sum_{i=1..n-1} a(i)*a(n-i). - Benoit Cloitre, Mar 16 2004
D-finite with recurrence a(n) = (5(2n-1)a(n-1) - 9(n-2)a(n-2))/(n+1) for n>=2; a(0)=a(1)=1. - Emeric Deutsch, Mar 20 2004
Moment representation: a(n)=(1/(8*Pi))*Int(x^n*sqrt(-x^2+10x-9)/x,x,1,9)+(3/4)*0^n. - Paul Barry, Sep 30 2009
a(n) = upper left term in M^n, M = the production matrix:
  1, 1
  4, 4, 4
  1, 1, 1, 1
  4, 4, 4, 4, 4
  1, 1, 1, 1, 1, 1
  ... - Gary W. Adamson, Jul 08 2011
a(n) is the sum of top row terms of Q^(n-1), where Q = the following infinite square production matrix:
  1, 4, 0, 0, 0, ...
  1, 1, 4, 0, 0, ...
  1, 1, 1, 4, 0, ...
  1, 1, 1, 1, 4, ...
  ... - Gary W. Adamson, Aug 23 2011
G.f.: (1+3*x-sqrt(9*x^2-10*x+1))/(8*x)=(1+3*x -G(0))/(4*x) ; G(k)= 1+x*3-x*4/G(k+1); (continued fraction, 1-step ). - Sergei N. Gladkovskii, Jan 05 2012
a(n) ~ sqrt(2)*3^(2*n+1)/(8*sqrt(Pi)*n^(3/2)). - Vaclav Kotesovec, Oct 11 2012
a(n) = A127846(n) for n>0. - Philippe Deléham, Apr 03 2013
0 = a(n)*(+81*a(n+1) - 225*a(n+2) + 36*a(n+3)) + a(n+1)*(+45*a(n+1) + 82*a(n+2) - 25*a(n+3)) + a(n+2)*(+5*a(n+2) + a(n+3)) for all n>=0. - Michael Somos, Aug 25 2014
G.f.: 1/(1 - x/(1 - 4*x/(1 - x/(1 - 4*x/(1 - x/(1 - ...)))))), a continued fraction. - Ilya Gutkovskiy, Aug 10 2017

A051485 Number of double nodes (exactly two nodes on that level) for all Motzkin paths of length n.

Original entry on oeis.org

0, 1, 1, 2, 6, 14, 32, 74, 180, 457, 1195, 3177, 8526, 23018, 62441, 170153, 465791, 1280956, 3538618, 9817619, 27348480, 76467497, 214532805, 603732396, 1703728554, 4819990947, 13667248631, 38834528740, 110556072877, 315290709729, 900635841754, 2576615923655, 7381956798465, 21177682172332, 60832837964492

Offset: 0

Wenjin Woan

nonn

			Of the 9 Motzkin paths of length 4 the following 5 have a total of 6 double nodes:
|1......|
|2../\..|3..__..|2.._...|2..._..|2......|
|2./..\.|2./..\.|3./.\_.|3._/.\.|3./\/\.|

Cf. A001006, A088457.

Column k=2 of A372014.

Edited by Michael Somos, Sep 29 2003

a(16)-a(34) from Alois P. Heinz, Apr 13 2024

cp's OEIS Frontend

User: Wenjin Woan

A131178 Non-plane increasing unary binary (0-1-2) trees where the nodes of outdegree 1 come in 2 colors.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

PARI

PARI

PARI

Formula

Extensions

A131638 Increasing binary trees having exactly two vertices with outdegree 1.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

Formula

Extensions

A125307 Number of increasing trees with branches of height 1.

Views

Author

Keywords

References

Links

Programs

Mathematica

Maxima

PARI

Formula

Extensions

A125062 Number of increasing trees with hills of height 1.

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Formula

Extensions

A079280 Number of log-concave paths of length n starting from the origin (0,0) with steps from {N=(0,1), E=(1,0) and S=(0,-1)} that stay in the second octant and never touch the line y=x except possibly at the beginning or the end.

Views

Author

Keywords

Comments

Formula

Extensions

A059435 Number of lattice paths in plane starting at (0,0) and ending at (n,n) with steps from {(i,j): i+j > 0, i, j >= 0} that never go below the line y = x.

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Formula

A059231 Number of different lattice paths running from (0,0) to (n,0) using steps from S = {(k,k) or (k,-k): k positive integer} that never go below the x-axis.

Views

Author

Keywords

Comments