A080934 -id:A080934 - cp's OEIS frontend

A080936 Triangle read by rows: T(n,k) is the number of Dyck paths of semilength n and height k (1 <= k <= n).

1, 1, 1, 1, 3, 1, 1, 7, 5, 1, 1, 15, 18, 7, 1, 1, 31, 57, 33, 9, 1, 1, 63, 169, 132, 52, 11, 1, 1, 127, 482, 484, 247, 75, 13, 1, 1, 255, 1341, 1684, 1053, 410, 102, 15, 1, 1, 511, 3669, 5661, 4199, 1975, 629, 133, 17, 1, 1, 1023, 9922, 18579, 16017, 8778, 3366, 912, 168, 19, 1

Offset: 1

Henry Bottomley, Feb 25 2003

Sum of entries in row n is A000108(n) (the Catalan numbers).
From Gus Wiseman, Nov 16 2022: (Start)
Also the number of unlabeled ordered rooted trees with n nodes and height k. For example, row n = 5 counts the following trees:
  (oooo)  ((o)oo)   (((o))o)  ((((o))))
          ((oo)o)   (((o)o))
          ((ooo))   (((oo)))
          (o(o)o)   ((o(o)))
          (o(oo))   (o((o)))
          (oo(o))
          ((o)(o))
(End)

			T(3,2)=3 because we have UUDDUD, UDUUDD, and UUDUDD, where U=(1,1) and D=(1,-1). The other two Dyck paths of semilength 3, UDUDUD and UUUDDD, have heights 1 and 3, respectively. - _Emeric Deutsch_, Jun 08 2011
Triangle starts:
  1;
  1,  1;
  1,  3,   1;
  1,  7,   5,   1;
  1, 15,  18,   7,  1;
  1, 31,  57,  33,  9,  1;
  1, 63, 169, 132, 52, 11, 1;

N. G. de Bruijn, D. E. Knuth, and S. O. Rice, The average height of planted plane trees, in: Graph Theory and Computing (ed. T. C. Read), Academic Press, New York, 1972, pp. 15-22.

Alois P. Heinz, Rows n = 1..141, flattened
Tomás Aguilar-Fraga, Jennifer Elder, Rebecca E. Garcia, Kimberly P. Hadaway, Pamela E. Harris, Kimberly J. Harry, Imhotep B. Hogan, Jakeyl Johnson, Jan Kretschmann, Kobe Lawson-Chavanu, J. Carlos Martínez Mori, Casandra D. Monroe, Daniel Quiñonez, Dirk Tolson III, and Dwight Anderson Williams II, Interval and L-interval Rational Parking Functions, arXiv:2311.14055 [math.CO], 2023. See p. 11.
FindStat - Combinatorial Statistic Finder, The height of a Dyck path
FindStat - Combinatorial Statistic Finder, The height of a Dyck path., The depth of an ordered tree., The depth minus 1 of an ordered tree
Anthony Joseph and Polyxeni Lamprou, A new interpretation of Catalan numbers, arXiv preprint arXiv:1512.00406 [math.CO], 2015.
Germain Kreweras, Sur les éventails de segments, Cahiers du Bureau Universitaire de Recherche Opérationelle, Cahier no. 15, Paris, 1970, pp. 3-41.
Germain Kreweras, Sur les éventails de segments, Cahiers du Bureau Universitaire de Recherche Opérationnelle, Institut de Statistique, Université de Paris, #15 (1970), 3-41. [Annotated scanned copy]
Kevin Limanta, Hopein Christofen Tang, and Yozef Tjandra, Permutation-generated maps between Dyck paths, arXiv:2105.14439 [math.CO], 2021. Mentions this sequence.
Dimana Miroslavova Pramatarova, Investigating the Periodicity of Weighted Catalan Numbers and Generalizing Them to Higher Dimensions, MIT Res. Sci. Instit. (2025). See p. 13.
Thomas Tunstall, Tim Rogers, and Wolfram Möbius, Assisted percolation of slow-spreading mutants in heterogeneous environments, arXiv:2303.01579 [q-bio.PE], 2023.
Vince White, Enumeration of Lattice Paths with Restrictions, (2024). Electronic Theses and Dissertations. 2799. See pp. 19, 25.

Cf. A000108 (row sums), A079214, A080934, A080935, A259885, A259899, A289481.
Columns k=1-10 give: A000012, A000225(n-1), A258109, A262600, A289418, A289419, A289420, A289421, A289422, A289423.
T(2n,n) gives A268316.
Counting by leaves instead of height gives A001263.
The unordered version is A034781.
The height statistic is ranked by A358379, unordered A109082.
Cf. A000081, A005043, A032027, A284778.

f := proc (k) options operator, arrow:
   sum(binomial(k-i, i)*(-z)^i, i = 0 .. floor((1/2)*k))
end proc:
h := proc (k) options operator, arrow:
   z^k/(f(k)*f(k+1))
end proc:
T := proc (n, k) options operator, arrow:
   coeff(series(h(k), z = 0, 25), z, n)
end proc:
for n to 11 do seq(T(n, k), k = 1 .. n) end do; # yields sequence in triangular form Emeric Deutsch, Jun 08 2011
# second Maple program:
b:= proc(x, y, k) option remember; `if`(y>min(k, x) or y<0, 0,
      `if`(x=0, 1, b(x-1, y-1, k)+ b(x-1, y+1, k)))
    end:
T:= (n, k)-> b(2*n, 0, k) -`if`(k=0, 0, b(2*n, 0, k-1)):
seq(seq(T(n, k), k=1..n), n=1..14);  # Alois P. Heinz, Aug 06 2012

b[x_, y_, k_] := b[x, y, k] = If[y > Min[k, x] || y<0, 0, If[x == 0, 1, b[x-1, y-1, k] + b[x-1, y+1, k]]]; T[n_, k_] := b[2*n, 0, k] - If[k == 0, 0, b[2*n, 0, k-1] ]; Table[Table[T[n, k], {k, 1, n}], {n, 1, 14}] // Flatten (* Jean-François Alcover, Feb 26 2015, after Alois P. Heinz *)
aot[n_]:=If[n==1,{{}},Join@@Table[Tuples[aot/@c],{c,Join@@Permutations/@IntegerPartitions[n-1]}]];
Table[Length[Select[aot[n],Depth[#]-2==k&]],{n,1,9},{k,1,n-1}] (* Gus Wiseman, Nov 16 2022 *)

T(n, k) = A080934(n, k) - A080934(n, k-1).
The g.f. for Dyck paths of height k is h(k) = z^k/(f(k)*f(k+1)), where f(k) are Fibonacci type polynomials defined by f(0)=f(1)=1, f(k)=f(k-1)-z*f(k-2) or by f(k) = Sum_{i=0..floor(k/2)} binomial(k-i,i)*(-z)^i. Incidentally, the g.f. for Dyck paths of height at most k is H(k) = f(k)/f(k+1). - Emeric Deutsch, Jun 08 2011
For all n >= 1 and floor((n+1)/2) <= k <= n we have: T(n,k) = 2*(2*k+3)*(2*k^2+6*k+1-3*n)*(2*n)!/((n-k)!*(n+k+3)!). - Gheorghe Coserea, Dec 06 2015
T(n, k) = Sum_{i=1..k-1} (-1)^(i+1) * (Sum_{j=1..n} (Sum_{x=0..n} (-1)^(j+x) * binomial(x+2n-2j+1,x))) * a(k-i); a(1)=1, a(0)=0. - Tim C. Flowers, May 14 2018

A080937 Number of Catalan paths (nonnegative, starting and ending at 0, step +/-1) of 2*n steps with all values <= 5.

Original entry on oeis.org

1, 1, 2, 5, 14, 42, 131, 417, 1341, 4334, 14041, 45542, 147798, 479779, 1557649, 5057369, 16420730, 53317085, 173118414, 562110290, 1825158051, 5926246929, 19242396629, 62479659622, 202870165265, 658715265222, 2138834994142, 6944753544643, 22549473023585

Offset: 0

Henry Bottomley, Feb 25 2003

With interpolated zeros (1,0,1,0,2,...), counts closed walks of length n at start or end node of P_6. The sequence (0,1,0,2,...) counts walks of length n between the start and second node. - Paul Barry, Jan 26 2005
HANKEL transform of sequence and the sequence omitting a(0) is the sequence A130716. This is the unique sequence with that property. - Michael Somos, May 04 2012
From Wolfdieter Lang, Mar 30 2020: (Start)
a(n) is also the upper left entry of the n-th power of the 3 X 3 tridiagonal matrix M_3 = Matrix([1,1,0], [1,2,1], [0,1,2]) from A332602: a(n) = ((M_3)^n)[1,1].
Proof: (M_3)^n = b(n-2)*(M_3)^2 - (6*b(n-3) - b(n-4))*M_3 + b(n-3)*1_3, for n >= 0, with b(n) = A005021(n), for n >= -4. For the proof of this see a comment in A005021. Hence (M_3)^n[1,1] = 2*b(n-2) - 5*b(n-3) + b(n-4), for n >= 0. This proves the 3 X 3 part of the conjecture in A332602 by Gary W. Adamson.
The formula for a(n) given below in terms of r = rho(7) = A160389 proves that a(n)/a(n-1) converges to rho(7)^2 = A116425 = 3.2469796..., because r - 2/r = 0.6920... < 1, and r^2 - 3 = 0.2469... < 1. This limit was conjectured in A332602 by Gary W. Adamson.
(End)

			G.f. = 1 + x + 2*x^2 + 5*x^3 + 14*x^4 + 42*x^5 + 131*x^6 + 417*x^7 + 1341*x^8 + ...

Alois P. Heinz, Table of n, a(n) for n = 0..600
Jean Luc Baril, Rigoberto Flórez, and José L. Ramirez, Generalized Narayana arrays, restricted Dyck paths, and related bijections, Univ. Bourgogne (France, 2025). See p. 27.
Jean-Luc Baril, Toufik Mansour, José L. Ramírez, and Mark Shattuck, Catalan words avoiding a pattern of length four, Univ. de Bourgogne (France, 2024). See p. 3.
Jean-Luc Baril and Helmut Prodinger, Enumeration of partial Lukasiewicz paths, arXiv:2205.01383 [math.CO], 2022.
Paul Barry, Centered polygon numbers, heptagons and nonagons, and the Robbins numbers, arXiv:2104.01644 [math.CO], 2021.
Giulio Cerbai, Anders Claesson, and Luca Ferrari, Stack sorting with restricted stacks, arXiv:1907.08142 [cs.DS], 2019.
Wei Chen, Enumeration of Set Partitions Refined by Crossing and Nesting Numbers, MS Thesis, Department of Mathematics. Simon Fraser University, Fall 2014. Table 4.1, line k=2.
Johann Cigler, Number of bounded Dyck paths with "negative length", MathOverflow question, Sep 26 2020.
Michael Dairyko, Lara Pudwell, Samantha Tyner, and Casey Wynn, Non-contiguous pattern avoidance in binary trees, Electron. J. Combin. 19 (2012), no. 3, Paper 22, 21 pp. MR2967227.
Nachum Dershowitz, Between Broadway and the Hudson: A Bijection of Corridor Paths, arXiv:2006.06516 [math.CO], 2020.
Paul Duncan and Einar Steingrimsson, Pattern avoidance in ascent sequences, arXiv preprint arXiv:1109.3641 [math.CO], 2011.
Sergi Elizalde, Symmetric peaks and symmetric valleys in Dyck paths, arXiv:2008.05669 [math.CO], 2020.
Stefan Felsner and Daniel Heldt, Lattice Path Enumeration and Toeplitz Matrices, J. Int. Seq. 18 (2015) # 15.1.3.
Daniel Heldt, On the mixing time of the face flip-and up/down Markov chain for some families of graphs, Dissertation, Mathematik und Naturwissenschaften der Technischen Universitat Berlin zur Erlangung des akademischen Grades Doktor der Naturwissenschaften, 2016.
Matthew Hyatt and Jeffrey Remmel, The classification of 231-avoiding permutations by descents and maximum drop, arXiv preprint arXiv:1208.1052 [math.CO], 2012. - From _N. J. A. Sloane_, Dec 24 2012
Aleksandar Ilic and Andreja Ilic, On the number of restricted Dyck paths, Filomat 25:3 (2011), 191-201; DOI: 10.2298/FIL1103191I.
Sergey Kitaev, Jeffrey Remmel and Mark Tiefenbruck, Marked mesh patterns in 132-avoiding permutations I, arXiv:1201.6243v1 [math.CO], 2012 (Corollary 3, case k=5, pages 10-11). - From _N. J. A. Sloane_, May 09 2012
Erkko Lehtonen and Tamás Waldhauser, Associative spectra of graph algebras I. Foundations, undirected graphs, antiassociative graphs, arXiv:2011.07621 [math.CO], 2020. See also J. of Algebraic Combinatorics (2021) Vol. 53, 613-638.
Toufik Mansour and Mark Shattuck, Some enumerative results related to ascent sequences, arXiv preprint arXiv:1207.3755 [math.CO], 2012. - From _N. J. A. Sloane_, Dec 22 2012
Sophie Morier-Genoud, Valentin Ovsienko, and Serge Tabachnikov, 2-frieze patterns and the cluster structure of the space of polygons, arXiv:1008.3359 [math.AG], 2010-2011. - From _N. J. A. Sloane_, Dec 26 2012
Sophie Morier-Genoud, Valentin Ovsienko, and Serge Tabachnikov, 2-frieze patterns and the cluster structure of the space of polygons, Annales de l'institut Fourier, 62 no. 3 (2012), 937-987. - From _N. J. A. Sloane_, Dec 26 2012
David Nečas and Ivan Ohlídal, Consolidated series for efficient calculation of the reflection and transmission in rough multilayers, Optics Express, Vol. 22, 2014, No. 4; DOI:10.1364/OE.22.004499.
László Németh and László Szalay, Sequences Involving Square Zig-Zag Shapes, J. Int. Seq., Vol. 24 (2021), Article 21.5.2.
Lara Pudwell, Pattern avoidance in trees, slides from a talk, mentions many sequences, 2012. - From _N. J. A. Sloane_, Jan 03 2013
Lara Pudwell and Andrew Baxter, Ascent sequences avoiding pairs of patterns, 2014.
Santiago Rojas-Rojas, Camila Muñoz, Edgar Barriga, Pablo Solano, Aldo Delgado, and Carla Hermann-Avigliano, Analytic Evolution for Complex Coupled Tight-Binding Models: Applications to Quantum Light Manipulation, arXiv:2310.12366 [quant-ph], 2023. See p. 12.
Index entries for linear recurrences with constant coefficients, signature (5,-6,1).

Cf. A000007, A000012, A001519, A007051, A011782, A024175, A080937, A080938.
Cf. A033191 which essentially provide the same sequence for different limits and tend to A000108.
Cf. A005021, A094790, A094789.
Cf. A211216, A005021, A160389, A116425, A332602.

I:=[1,1,2]; [n le 3 select I[n] else 5*Self(n-1)-6*Self(n-2)+Self(n-3): n in [1..30]]; // Vincenzo Librandi, Jan 09 2016

a:= n-> (<<0|1|0>, <0|0|1>, <1|-6|5>>^n. <<1, 1, 2>>)[1, 1]:
seq(a(n), n=0..35);  # Alois P. Heinz, Nov 09 2012

nn=56;Select[CoefficientList[Series[(1-4x^2+3x^4)/(1-5x^2+6x^4-x^6), {x,0,nn}], x],#>0 &] (* Geoffrey Critzer, Jan 26 2014 *)
LinearRecurrence[{5,-6,1},{1,1,2},30] (* Jean-François Alcover, Jan 09 2016 *)

a=vector(99); a[1]=1; a[2]=2;a[3]=5; for(n=4,#a,a[n]=5*a[n-1]-6*a[n-2] +a[n-3]); a \\ Charles R Greathouse IV, Jun 10 2011

{a(n) = if( n<0, n = -n; polcoeff( (1 - 3*x + x^2) / (1 - 6*x + 5*x^2 - x^3) + x * O(x^n), n), polcoeff( (1 - 4*x + 3*x^2) / (1 - 5*x + 6*x^2 - x^3) + x * O(x^n), n))} /* Michael Somos, May 04 2012 */

a(n) = A080934(n,5).
G.f.: (1-4*x+3*x^2)/(1-5*x+6*x^2-x^3). - Ralf Stephan, May 13 2003
a(n) = 5*a(n-1) - 6*a(n-2) + a(n-3). - Herbert Kociemba, Jun 11 2004
a(n) = A096976(2*n). - Floor van Lamoen, Nov 02 2005
a(n) = (4/7-4/7*cos(1/7*Pi)^2)*(4*(cos(Pi/7))^2)^n + (1/7-2/7*cos(1/7*Pi) + 4/7*cos(1/7*Pi)^2)*(4*(cos(2*Pi/7))^2)^n + (2/7+2/7*cos(1/7*Pi))*(4*(cos(3*Pi/7))^2)^n for n>=0. - Richard Choulet, Apr 19 2010
G.f.: 1 / (1 - x / (1 - x / (1 - x / (1 - x / (1 - x))))). - Michael Somos, May 04 2012
a(-n) = A038213(n). a(n + 2) * a(n) - a(n + 1)^2 = a(1 - n). Convolution inverse is A123183 with A123183(0)=1. - Michael Somos, May 04 2012
From Wolfdieter Lang, Mar 30 2020: (Start)
In terms of the algebraic number r = rho(7) = A160389 of degree 3 the formula given by Richard Choulet becomes a(n) = (1/7)*(r)^(2*n)*(C1(r) + C2(r)*(r - 2/r)^(2*n) + C3(r)*(r^2 - 3)^(2*n)), with C1(r) = 4 - r^2, C2(r) = 1 - r + r^2, and C3 = 2 + r.
a(n) = ((M_3)^n)[1,1] = 2*b(n-2) - 5*b(n-3) + b(n-4), for n >= 0, with the 3 X 3 tridiagonal matrix M_3 = Matrix([1,1,0], [1,2,1], [0,1,2]) from A332602, and b(n) = A005021(n) (with offset n >= -4). (End)

A211216 Expansion of (1-8x+21x^2-20x^3+5x^4)/(1-9x+28x^2-35x^3+15x^4-x^5).

Original entry on oeis.org

1, 1, 2, 5, 14, 42, 132, 429, 1430, 4862, 16795, 58766, 207783, 740924, 2660139, 9603089, 34818270, 126676726, 462125928, 1689438278, 6186432967, 22682699779, 83249302471, 305773834030, 1123771473120, 4131947428007, 15197952958467, 55915691993228

Offset: 0

Bruno Berselli, May 11 2012

In the paper of Kitaev, Remmel and Tiefenbruck (see the Links section), Q_(132)^(k,0,0,0)(x,0) represents a generating function depending on k and x.
For successive values of k we have:
k=1, the g.f. of A000012: 1/(1-x);
k=2,      "      A011782: (1-x)/(1-2*x);
k=3,      "      A001519: (1-2*x)/(1-3*x+x^2);
k=4,      "      A124302: (1-3*x+x^2)/(1-4*x+3*x^2);
k=5,      "      A080937: (1-4*x+3*x^2)/(1-5*x+6*x^2-x^3);
k=6,      "      A024175: (1-5*x+6*x^2-x^3)/(1-6*x+10*x^2-4*x^3);
k=7,      "      A080938: (1-6*x+10*x^2-4*x^3)/(1-7*x+15*x^2-10*x^3+x^4);
k=8,      "      A033191: (1-7*x+15*x^2-10*x^3+x^4)/(1-8*x+21*x^2
                           -20*x^3+5*x^4).
This sequence corresponds to the case k=9.
We observe that the coefficients of numerators and denominators are in A115139.
In general, Q_(132)^(k,0,0,0)(x,0) is the generating function for Dyck paths whose maximum height is less than or equal to k; also, it is the generating function of rooted binary trees T which have no nodes 'eta' such that there are >= k left edges on the path from 'eta' to the root of T (see cited paper, page 11).

Bruno Berselli, Table of n, a(n) for n = 0..1000
Sergey Kitaev, Jeffrey Remmel and Mark Tiefenbruck, Marked mesh patterns in 132-avoiding permutations I, arXiv:1201.6243v1 [math.CO], 2012 (page 10, Corollary 3).
K. Mészáros, A. H. Morales, and J. Striker, On flow polytopes, order polytopes, and certain faces of the alternating sign matrix polytope, arXiv preprint arXiv:1510.03357 [math.CO], 2015-2019.
László Németh and László Szalay, Sequences Involving Square Zig-Zag Shapes, J. Int. Seq., Vol. 24 (2021), Article 21.5.2.
Index entries for linear recurrences with constant coefficients, signature (9,-28,35,-15,1).

Cf. A000108, A001519, A011782, A024175, A033191, A080937, A080938, A124302.

Cf. square array in A080934.

m:=28; R:=PowerSeriesRing(Integers(), m); Coefficients(R!((1-8*x+21*x^2-20*x^3+5*x^4)/(1-9*x+28*x^2-35*x^3+15*x^4-x^5)));

CoefficientList[Series[(1 - 8 x + 21 x^2 - 20 x^3 + 5 x^4)/(1 - 9 x + 28 x^2 - 35 x^3 + 15 x^4 - x^5), {x, 0, 27}], x]

Vec((1-8*x+21*x^2-20*x^3+5*x^4)/(1-9*x+28*x^2-35*x^3+15*x^4-x^5)+O(x^28))

G.f.: (1-3*x+x^2)*(1-5*x+5*x^2)/(1-9*x+28*x^2-35*x^3+15*x^4-x^5).
G.f.: 1/(1-x/(1-x/(1-x/(1-x/(1-x/(1-x/(1-x/(1-x/(1-x))))))))). - Philippe Deléham, Mar 14 2013
a(n) = A000108(n) + Sum_{k=1..n} (4*binomial(2*n, n+11*k) - binomial(2*n+2, n+11*k+1)). - Greg Dresden, Jan 28 2023

A094718 Array T read by antidiagonals: T(n,k) is the number of involutions avoiding 132 and 12...k.

Original entry on oeis.org

0, 1, 0, 1, 1, 0, 1, 2, 1, 0, 1, 2, 2, 1, 0, 1, 2, 3, 4, 1, 0, 1, 2, 3, 5, 4, 1, 0, 1, 2, 3, 6, 8, 8, 1, 0, 1, 2, 3, 6, 9, 13, 8, 1, 0, 1, 2, 3, 6, 10, 18, 21, 16, 1, 0, 1, 2, 3, 6, 10, 19, 27, 34, 16, 1, 0, 1, 2, 3, 6, 10, 20, 33, 54, 55, 32, 1, 0, 1, 2, 3, 6, 10, 20, 34, 61, 81, 89, 32, 1

Offset: 1

Ralf Stephan, May 23 2004

Also, number of paths along a corridor with width k, starting from one side (from H. Bottomley's comment in A061551).
Rows converge to binomial(n,floor(n/2)) (A001405).
Note that the rows and columns start at 1, which for example obscures the fact that the first row refers to A000007 and not to A000004. A better choice is the indexing 0 <= k and 0 <= n. The Maple program below uses this indexing and builds only on the roots of -1. - Peter Luschny, Sep 17 2020

			Array begins
  0   0   0   0   0   0   0   0   0   0 ...
  1   1   1   1   1   1   1   1   1   1 ...
  1   2   2   4   4   8   8  16  16  32 ...
  1   2   3   5   8  13  21  34  55  89 ...
  1   2   3   6   9  18  27  54  81 162 ...
  1   2   3   6  10  19  33  61 108 197 ...
  1   2   3   6  10  20  34  68 116 232 ...
  1   2   3   6  10  20  35  69 124 241 ...
  1   2   3   6  10  20  35  70 125 250 ...
  1   2   3   6  10  20  35  70 126 251 ...
  ...

Shaun V. Ault and Charles Kicey, Counting paths in corridors using circular Pascal arrays, Discrete Mathematics, Volume 332, October 2014, Pages 45-54.
Nachum Dershowitz, Between Broadway and the Hudson, arXiv:2006.06516 [math.CO], 2020.
Robert Dutton Fray and David Paul Roselle, Weighted lattice paths, Pacific Journal of Mathematics, 37(1) (1971), 85-96.
O. Guibert and T. Mansour, Restricted 132-involutions, Séminaire Lotharingien de Combinatoire, B48a (2002), 23 pp.
T. Mansour, Restricted even permutations and Chebyshev polynomials, arXiv:math/0302014 [math.CO], 2003.

Rows 3-8 are A016116, A000045, A038754, A028495, A030436, A061551.
Main diagonal is A014495, antidiagonal sums are in A094719.
Cf. A080934 (permutations).

X := (j, n) -> (-1)^(j/(n+1)) - (-1)^((n-j+1)/(n+1)):
R := n -> select(k -> type(k, odd), [$(1..n)]):
T := (n, k) -> add((2 + X(j,n))*X(j,n)^k, j in R(n))/(n+1):
seq(lprint([n], seq(simplify(T(n, k)), k=0..10)), n=0..9); # Peter Luschny, Sep 17 2020

U = ChebyshevU;
m = maxExponent = 14;
row[1] = Array[0&, m];
row[k_] := 1/(x U[k, 1/(2x)])*Sum[U[j, 1/(2x)], {j, 0, k-1}] + O[x]^m // CoefficientList[#, x]& // Rest;
T = Table[row[n], {n, 1, m}];
Table[T[[n-k+1, k]], {n, 1, m-1}, {k, 1, n}] // Flatten (* Jean-François Alcover, Nov 17 2018 *)

G.f. for k-th row: 1/(x*U(k, 1/(2*x))) * Sum_{j=0..k-1} U(j, 1/(2*x)), with U(k, x) the Chebyshev polynomials of second kind. [Clarified by Jean-François Alcover, Nov 17 2018]

T(n, k) = (1/(n+1))*Sum_{j=1..n, j odd} (2 + [j, n]) * [j, n]^k where [j, n] := (-1)^(j/(n+1)) - (-1)^((n-j+1)/(n+1)). - Peter Luschny, Sep 17 2020

A080938 Number of Catalan paths (nonnegative, starting and ending at 0, step +-1) of 2*n steps with all values less than or equal to 7.

Original entry on oeis.org

1, 1, 2, 5, 14, 42, 132, 429, 1429, 4846, 16645, 57686, 201158, 704420, 2473785, 8704089, 30664890, 108126325, 381478030, 1346396146, 4753200932, 16783118309, 59266297613, 209302921830, 739203970773, 2610763825782, 9221050139566, 32568630376132

Offset: 0

Henry Bottomley, Feb 25 2003

From Wolfdieter Lang, Mar 27 2020: (Start)
a(n) also gives the upper left entry of the n-th power of the 4 X 4 tridiagonal matrix M_4, given in A332602: M_4 = Matrix([1,1,0,0], [1,2,1,0], [0,1,2,1], [0,0,1,2]): a(n) = (M_4)^n[1,1]. Proof from the formula for (M_4)^n, given in a comment in A094256, derived from the Cayley-Hamilton theorem, which leads to the recurrence. The formula for a(n) in terms of A094256 is given below.
For A094256(n+1)/A094256(n), like for A094829(n+1)/A094829(n), the limit for n -> infinity is rho(9)^2 = A332438 = 3.53208888..., with rho(9) = 2*cos(Pi/9) = A332437. Therefore the formula of a(n) in terms of A094256 shows that the same limit is reached for a(n+1)/a(n). See this conjecture by Gary W. Adamson in A332602.
(End)

			1 + x + 2*x^2 + 5*x^3 + 14*x^4 + 42*x^5 + 132*x^6 + 429*x^7 + ...

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Jean-Luc Baril, Toufik Mansour, José L. Ramírez, and Mark Shattuck, Catalan words avoiding a pattern of length four, Univ. de Bourgogne (France, 2024). See p. 3.
Wei Chen, Enumeration of Set Partitions Refined by Crossing and Nesting Numbers, MS Thesis, Department of Mathematics. Simon Fraser University, Fall 2014. Table 4.1, k=4.
Michael Dairyko, Lara Pudwell, Samantha Tyner, and Casey Wynn, Non-contiguous pattern avoidance in binary trees. Electron. J. Combin. 19 (2012), no. 3, Paper 22, 21 pp. MR2967227.
Sergey Kitaev, Jeffrey Remmel and Mark Tiefenbruck, Marked mesh patterns in 132-avoiding permutations I, arXiv:1201.6243v1 [math.CO], 2012 (Corollary 3, case k=7, pages 10-11). - From _N. J. A. Sloane_, May 09 2012.
László Németh and László Szalay, Sequences Involving Square Zig-Zag Shapes, J. Int. Seq., Vol. 24 (2021), Article 21.5.2.
Lara Pudwell, Pattern avoidance in trees (slides from a talk, mentions many sequences), 2012.
Index entries for linear recurrences with constant coefficients, signature (7,-15,10,-1).

Cf. A000007, A000012, A011782, A001519, A007051, A080937, A024175, A033191 which essentially provide the same sequence for different limits and tend to A000108.

Cf. A211216, A094826 (first differences), A094829, A094829, A332602, A332437, A332438.

I:=[1,1,2,5]; [n le 4 select I[n] else 7*Self(n-1)-15*Self(n-2)+10*Self(n-3)-Self(n-4): n in [1..30]]; // Vincenzo Librandi, Nov 30 2018

CoefficientList[Series[(1 - 2 x) (2 x^2 - 4 x + 1) / ((x - 1) (x^3 - 9 x^2 + 6 x - 1)), {x, 0, 40}], x] (* Vincenzo Librandi, Nov 30 2018 *)
LinearRecurrence[{7, -15, 10, -1}, {1, 1, 2, 5}, 30] (* Jean-François Alcover, Jan 07 2019 *)

{a(n) = local(A); A = 1; for( i=1, 7, A = 1 / (1 - x*A)); polcoeff( A + x * O(x^n), n)} /* Michael Somos, May 12 2012 */

a(n) = A080934(n,7).
G.f.: -(2*x - 1)*(2*x^2 - 4*x + 1) / ( (x - 1)*(x^3 - 9*x^2 + 6*x - 1) ). - Ralf Stephan, May 13 2003
a(n) = 7*a(n-1) - 15*a(n-2) + 10*a(n-3) - a(n-4). - Herbert Kociemba, Jun 13 2004
G.f.: 1 / (1 - x / (1 - x / (1 - x / (1 - x / (1 - x / (1 - x / (1 - x))))))). - Michael Somos, May 12 2012
a(n) = 5*b(n-2) - 21*b(n-3) + 19*b(n-4) - 2*b(n-5), for n >= 0, with b(n) = A094256(n), for n >= -5. See a comment in A094256 for this offset, and the above comment. - Wolfdieter Lang, Mar 28 2020

A083935 Inverse function of N -> N injection A083934.

Original entry on oeis.org

0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 3, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0

Offset: 0

Antti Karttunen, May 13 2003

nonn

a(0)=0 because A083934(0)=0, but a(n) = 0 also for those n which do not occur as values of A083934. All positive natural numbers occur here once.

a(A080934(n)) = n for all n. Cf. A083925-A083929, A014486, A080300, A059905, A059906.

A013699 Degree of variety K_{2,n}^2.

Original entry on oeis.org

1, 32, 610, 9842, 147798, 2145600, 30664890, 435668420, 6186432967, 88066807556, 1258885297696, 18084694597452, 261164661944060, 3791317346771584, 55316720239735242, 810944384733610356

Offset: 1

Joachim.Rosenthal(AT)nd.edu (Joachim Rosenthal)

Number of Catalan paths (nonnegative, starting and ending at 0, step +/-1) of 4n+4 steps with all values less than or equal to n+1 (see A080934).

M. S. Ravi et al., Dynamic pole assignment and Schubert calculus, SIAM J. Control Optimization, 34 (1996), 813-832, esp. p. 825.

Cf. A013698 (q=1), A013700 (q=3), A013701 (q=4), A013702 (q=5).

K(n,q=2)=(2*n+n*q+2*q)!*sum(j=0,q,((q-2*j)*(n+2)+1)/(n+j*(n+2))!/(n+1+(q-j)*(n+2))!)

A013701 Degree of variety K_{2,n}^4.

Original entry on oeis.org

1, 512, 75025, 7174454, 562110290, 39541748736, 2610763825782, 165745451110910, 10262482704258873, 625250747214775916, 37701606156514031251, 2258713106034310399852, 134810129909509070121060

Offset: 1

Joachim.Rosenthal(AT)nd.edu (Joachim Rosenthal)

Number of Catalan paths (nonnegative, starting and ending at 0, step +/-1) of 6n+8 steps with all values less than or equal to n+1 (see A080934).

M. S. Ravi et al., Dynamic pole assignment and Schubert calculus, SIAM J. Control Optimization, 34 (1996), 813-832, esp. p. 825.

Cf. A013698 (q=1), A013699 (q=2), A013700 (q=3), A013702 (q=5).

K(n,q=4)=(2*n+n*q+2*q)!*sum(j=0,q,((q-2*j)*(n+2)+1)/(n+j*(n+2))!/(n+1+(q-j)*(n+2))!)

A080935 Triangle read by rows of number of Catalan paths (nonnegative, starting and ending at 0, step +/-1) of 2n steps with all values less than or equal to k.

Original entry on oeis.org

1, 1, 2, 1, 4, 5, 1, 8, 13, 14, 1, 16, 34, 41, 42, 1, 32, 89, 122, 131, 132, 1, 64, 233, 365, 417, 428, 429, 1, 128, 610, 1094, 1341, 1416, 1429, 1430, 1, 256, 1597, 3281, 4334, 4744, 4846, 4861, 4862, 1, 512, 4181, 9842, 14041, 16016, 16645, 16778, 16795

Offset: 1

Henry Bottomley, Feb 25 2003

T(n,k) is the number of different out-stack sequences of n elements to be pushed into a stack of size k. E.g. T(3,2) = 4 since the 4 possible out-stack sequences are 123, 132, 213, 231; 321 is not allowed since it requires a stack of size 3. - Jianing Song, Oct 28 2021

			Rows start:
  1;
  1,2;
  1,4,5;
  1,8,13,14;
  1,16,34,41,42;
  ...
T(3,2)=4 since the paths of length 2*3 (7 points) with all values less than or equal to 2 can take the routes 0101010, 0101210, 0121010 or 0121210, but not 0123210.

Vince White, Enumeration of Lattice Paths with Restrictions, (2024). Electronic Theses and Dissertations. 2799. See pp. 20, 25.

Cf. A000108, A079214, A080934, A080936.

For 1<=k<=n, T(n, k) =A080934(n, k) =T(n, k-1)+A080936(n, k).

A082635 Square array read by antidiagonals: degree of the K(2,p)^q variety.

Original entry on oeis.org

1, 2, 1, 5, 8, 1, 14, 55, 32, 1, 42, 364, 610, 128, 1, 132, 2380, 9842, 6765, 512, 1, 429, 15504, 147798, 265720, 75025, 2048, 1, 1430, 100947, 2145600, 9112264, 7174454, 832040, 8192, 1, 4862, 657800, 30664890, 290926848, 562110290, 193710244

Offset: 1

Ralf Stephan, May 14 2003

Numbers are related to the dynamic pole assignment problem. "The variety K(m,p)^q can also be viewed as the parameterization of the space of rational curves of degree q of the Grassmann variety Grass(m,m+p)".

Also lim(n->inf, T(n+1,2i)/T(n,2i)) = 4^(i+1).

			Top left corner of array:
1,2,5,14,42,132,429,1430,... A000108 (Catalan numbers)
1,8,55,364,2380,15504,100947,...A013068 deg K(2,n)^1
1,32,610,9842,147798,2145600,...A013069 deg K(2,n)^2
1,128,6765,265720,9112264,... A013070 deg K(2,n)^3
1,512,75025,7174454,... A013071 deg K(2,n)^4

M. S. Ravi et al., Dynamic pole assignment and Schubert calculus, SIAM J. Control Optimization, 34 (1996), 813-832, esp. p. 825.

Cf. A013702.
Second column is A004171(q), third is A000045(5q).
T(n, 2i) = A080934((i+1)n+2i, n+1).

degK2(p, q)=(-1)^q*(2p+pq+2q)!*sum(j=0, q, ((q-2j)(p+2)+1)/(p+j(p+2))!/(p+1+(q-j)(p+2))!).

cp's OEIS Frontend

A080936 Triangle read by rows: T(n,k) is the number of Dyck paths of semilength n and height k (1 <= k <= n).

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A080937 Number of Catalan paths (nonnegative, starting and ending at 0, step +/-1) of 2*n steps with all values <= 5.

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A211216 Expansion of (1-8*x+21*x^2-20*x^3+5*x^4)/(1-9*x+28*x^2-35*x^3+15*x^4-x^5).

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A094718 Array T read by antidiagonals: T(n,k) is the number of involutions avoiding 132 and 12...k.

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A080938 Number of Catalan paths (nonnegative, starting and ending at 0, step +-1) of 2*n steps with all values less than or equal to 7.

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A083935 Inverse function of N -> N injection A083934.

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A013699 Degree of variety K_{2,n}^2.

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A211216 Expansion of (1-8x+21x^2-20x^3+5x^4)/(1-9x+28x^2-35x^3+15x^4-x^5).