A262600 -id:A262600 - 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

A258109 Number of balanced parenthesis expressions of length 2n and depth 3.

Original entry on oeis.org

1, 5, 18, 57, 169, 482, 1341, 3669, 9922, 26609, 70929, 188226, 497845, 1313501, 3459042, 9096393, 23895673, 62721698, 164531565, 431397285, 1130708866, 2962826465, 7761964833, 20331456642, 53249182309, 139449644717, 365166860706, 956185155129, 2503657040137

Offset: 3

Gheorghe Coserea, May 20 2015

a(n) is the number of Dyck paths of length 2n and height 3. For example, a(3) = 1 because there is only one such Dyck path which is UUUDDD. - Ran Pan, Sep 26 2015

a(n) is the number of rooted plane trees with n+1 nodes and height 3 (see example for correspondence). - Gheorghe Coserea, Feb 04 2016

			For n=4, the a(4) = 5 solutions are
                /\       /\
               /  \        \
LRLLLRRR    /\/    \        \
................................
                /\        |
             /\/  \      / \
LLRLLRRR    /      \        \
................................
              /\/\        |
             /    \       |
LLLRLRRR    /      \     / \
................................
              /\          |
             /  \/\      / \
LLLRRLRR    /      \    /
................................
              /\          /\
             /  \        /
LLLRRRLR    /    \/\    /

S. S. Skiena and M. A. Revilla, Programming Challenges: The Programming Contest Training Manual, Springer, 2006, page 140.

Alois P. Heinz, Table of n, a(n) for n = 3..1000 (first 40 terms from Gheorghe Coserea)
Kyu-Hwan Lee, Se-jin Oh, Catalan triangle numbers and binomial coefficients, arXiv:1601.06685 [math.CO], 2016.
Index entries for linear recurrences with constant coefficients, signature (5,-7,2).

Column k=3 of A080936.
Column k=2 of A287213.
Cf. A000045, A000079, A262600.

typedef long long unsigned Integer;
Integer a(int n)
{
    int i;
    Integer pow2 = 1, a[3] = {0};
    for (i = 3; i <= n; ++i) {
        a[ i%3 ] = pow2 + 3 * a[ (i-1)%3 ] - a[ (i-2)%3 ];
        pow2 = pow2 * 2;
    }
    return a[ (i-1)%3 ];
}

I:=[1,5,18,57,169]; [n le 5 select I[n] else 5*Self(n-1) - 7*Self(n-2) + 2*Self(n-3): n in [1..40]]; // Vincenzo Librandi, Sep 26 2015

a:= proc(n) option remember; `if`(n<3, 0,
      `if`(n=3, 1, 2^(n-3) +3*a(n-1) -a(n-2)))
    end:
seq(a(n), n=3..30);  # Alois P. Heinz, May 20 2015

Join[{1, 5}, LinearRecurrence[{5, -7, 2}, {18, 57, 169}, 30]] (* Vincenzo Librandi, Sep 26 2015 *)

Vec(-x^3/((2*x-1)*(x^2-3*x+1)) + O(x^100)) \\ Colin Barker, May 24 2015

a(n) = fibonacci(2*n-1) - 2^(n-1)  \\ Gheorghe Coserea, Feb 04 2016

a(n) = 2^(n-3) + 3 * a(n-1) - a(n-2).
a(n) = 5*a(n-1) - 7*a(n-2) + 2*a(n-3) for n>5. - Colin Barker, May 24 2015
G.f.: -x^3 / ((2*x-1)*(x^2-3*x+1)). - Colin Barker, May 24 2015
a(n) = A000045(2n-1) - A000079(n-1). - Gheorghe Coserea, Feb 04 2016
a(n) = 2^(-1-n)*(-5*4^n - (-5+sqrt(5))*(3+sqrt(5))^n + (3-sqrt(5))^n*(5+sqrt(5))) / 5. - Colin Barker, Jun 05 2017
a(n) = Sum_{i=1..n-1} A061667(i)*(n-1-i) - Tim C. Flowers, May 16 2018

A285197 Expansion of (1-6x+11x^2-5x^3) / ((1-x)(1-3x)(1-3*x+x^2)).

Original entry on oeis.org

1, 1, 2, 6, 20, 67, 221, 717, 2294, 7258, 22760, 70863, 219353, 675769, 2073674, 6342414, 19345052, 58867195, 178779893, 542042565, 1641058046, 4962262306, 14989121072, 45235277511, 136407241265, 411058035697, 1237981634066, 3726531171222, 11212544793764, 33723901952563

Offset: 0

N. J. A. Sloane, Apr 30 2017

Colin Barker, Table of n, a(n) for n = 0..1000
M. H. Albert, M. D. Atkinson, and V. Vatter, Inflations of geometric grid classes: three case studies, arXiv:1209.0425 [math.CO], 2012.
Index entries for linear recurrences with constant coefficients, signature (7,-16,13,-3).

Cf. A262600.

LinearRecurrence[{7,-16,13,-3},{1,1,2,6},30] (* Harvey P. Dale, Apr 01 2018 *)

Vec((1-6*x+11*x^2-5*x^3) / ((1-x)*(1-3*x)*(1-3*x+x^2)) + O(x^30)) \\ Colin Barker, May 01 2017

From Colin Barker, May 01 2017: (Start)
a(n) = 7*a(n-1) - 16*a(n-2) + 13*a(n-3) - 3*a(n-4) for n>3.
a(n) = (1/2 + 3^n/2 + (2^(-n)*((3-sqrt(5))^n - (3+sqrt(5))^n)) / sqrt(5)).
(End)
2*a(n) = 1 +3^n -2*A001906(n). - R. J. Mathar, Aug 19 2022

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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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A258109 Number of balanced parenthesis expressions of length 2n and depth 3.

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A285197 Expansion of (1-6*x+11*x^2-5*x^3) / ((1-x)*(1-3*x)*(1-3*x+x^2)).

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A285197 Expansion of (1-6x+11x^2-5x^3) / ((1-x)(1-3x)(1-3*x+x^2)).