A253180 -id:A253180 - cp's OEIS frontend

A002740 Number of tree-rooted bridgeless planar maps with two vertices and n faces.

0, 0, 0, 2, 15, 84, 420, 1980, 9009, 40040, 175032, 755820, 3233230, 13728792, 57946200, 243374040, 1017958725, 4242920400, 17631691440, 73078721100, 302202005490, 1247182879800, 5137916074200, 21132472200840, 86794082253450, 356013544661424, 1458583920435600, 5969389748449400

Offset: 0

N. J. A. Sloane

a(n) is the sum of the major indices of all Dyck words of length 2n-2. The major index of a Dyck word is the sum of the positions of those 1's that are followed by a 0. Example: a(4)=15 because the Dyck words of length 6 are 010101, 010011, 001101, 001011 and 000111 having major indices 6,2,4,3 and 0, respectively. a(n) = Sum_{k=0..n(n-1)} k*A129175(n,k). - Emeric Deutsch, Apr 20 2007

J. Ser, Les Calculs Formels des Séries de Factorielles, Gauthier-Villars, Paris, 1933, p. 97.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Mireille Bousquet-Mélou, New enumerative results on two-dimensional directed animals, Discr. Math., Vol. 180, No. 1-3 (1998), pp. 73-106. See Cor. 6.6.
Luca Ferrari and Emanuele Munarini, Enumeration of saturated chains in Dyck lattices, arXiv preprint arXiv:1203.6807 [math.CO], 2012.
J. Fürlinger and J. Hofbauer, q-Catalan numbers, J. Comb. Theory, Series A, Vol. 40, No. 2 (1985), pp. 248-264.
Clemens Heuberger, Sarah J. Selkirk, and Stephan Wagner, Enumeration of Generalized Dyck Paths Based on the Height of Down-Steps Modulo k, arXiv:2204.14023 [math.CO], 2022.
J. Ser, Les Calculs Formels des Séries de Factorielles, Gauthier-Villars, Paris, 1933 [Local copy].
J. Ser, Les Calculs Formels des Séries de Factorielles, Gauthier-Villars, Paris, 1933. (Annotated scans of some selected pages)
Mark Shattuck, Parity theorems for statistics on permutations and Catalan words, INTEGERS, Electronic J. of Combinatorial Number Theory, Vol. 5 (2005), Paper A07.
T. R. S. Walsh and A. B. Lehman, Counting rooted maps by genus III: Nonseparable maps, J. Combinatorial Theory Ser. B, Vol. 18, No. 3 (1975), pp. 222-259. See Table VIIIb.

Cf. A129175.

A diagonal of A253180.

[(n-2)*Binomial(2*n-2,n-2)/2 : n in [0..30]]; // Wesley Ivan Hurt, Sep 24 2014

with(combinat):for n from 0 to 22 do printf(`%d, `,n*sum(binomial(2*n, n)/(n+1)/2, k=2..n)) od: # Zerinvary Lajos, Mar 13 2007
a:=n->sum(sum(binomial(2*n,n)/(n+1)/2, j=1..n),k=2..n): seq(a(n), n=0..25); # Zerinvary Lajos, May 09 2007
A002740:=n->(n-2)*binomial(2*n-2,n-2)/2+0^n: seq(A002740(n), n=0..30); # Wesley Ivan Hurt, Sep 24 2014

a[n_] := (n-1)(n-2)Binomial[2(n-1), n-1]/(2n); a[0] = 0; Table[a[n], {n, 0, 23}] (* Jean-François Alcover, Nov 16 2011 *)

combinat::catalan(n) *binomial(n,2) $ n = 0..22 // Zerinvary Lajos, Feb 15 2007

a(n)=if(n<3,0,(2*(n-1))!/(2*n!*(n-3)!)); /* Joerg Arndt, Sep 28 2012 */

G.f.: (1/2)*(1-(1 - 6*t + 6*t^2)/(1-4*t)^(3/2)).
a(n+3) = (2*(n+2))!/(2*n!*(n+3)!). - Wolfdieter Lang
a(n+2) = Sum_{k=0..n} k*binomial(k+n, k). - Benoit Cloitre, Oct 25 2003
a(n) = Sum_{k=2..n} Sum_{j=1..n} binomial(2*n,n)/(2*(n+1)), n >= 0. - Zerinvary Lajos, May 09 2007
a(n) = (n-2)*binomial(2n-2, n-2)/2 + 0^n. - Wesley Ivan Hurt, Sep 24 2014
E.g.f.: (1 + exp(2*x) * ((2*x - 1) * BesselI(0,2*x) - x * BesselI(1,2*x))) / 2. - Ilya Gutkovskiy, Nov 03 2021
From Amiram Eldar, Mar 22 2022: (Start)
Sum_{n>=3} 1/a(n) = 3 - 4*Pi/(3*sqrt(3)).
Sum_{n>=3} (-1)^(n+1)/a(n) = 16*log(phi)/sqrt(5) - 3, where phi is the golden ratio (A001622). (End)

Name clarified by Noam Zeilberger, Aug 18 2017

A290605 Square array A(n,k), n>=0, k>=0, read by antidiagonals, where column k is the expansion of 2/(1 + sqrt(1 - 4kx)).

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 1, 2, 2, 0, 1, 3, 8, 5, 0, 1, 4, 18, 40, 14, 0, 1, 5, 32, 135, 224, 42, 0, 1, 6, 50, 320, 1134, 1344, 132, 0, 1, 7, 72, 625, 3584, 10206, 8448, 429, 0, 1, 8, 98, 1080, 8750, 43008, 96228, 54912, 1430, 0, 1, 9, 128, 1715, 18144, 131250, 540672, 938223, 366080, 4862, 0

Offset: 0

Ilya Gutkovskiy, Aug 07 2017

Number of 2n-length strings of balanced parentheses of at most k different types. Also number of binary trees with n inner nodes of at most k different dimensions. - Alois P. Heinz, Oct 28 2019

			G.f. of column k: A(x) = 1 + k*x + 2*k^2*x^2 + 5*k^3*x^3 + 14*k^4*x^4 + 42*k^5*x^5 + 132*k^6*x^6 + ...
Square array begins:
  1,   1,     1,      1,      1,       1,  ...
  0,   1,     2,      3,      4,       5,  ...
  0,   2,     8,     18,     32,      50,  ...
  0,   5,    40,    135,    320,     625,  ...
  0,  14,   224,   1134,   3584,    8750,  ...
  0,  42,  1344,  10206,  43008,  131250,  ...

Alois P. Heinz, Antidiagonals n = 0..140, flattened

Columns k=0-10 give:  A000007, A000108, A151374, A005159, A151403, A156058, A156128, A156266, A156270, A156273, A156275.
Rows n=0-2 give: A000012, A001477, A001105.
Main diagonal gives A291699.
Cf. A000108, A253180, A256061.

ctln:= proc(n) option remember; binomial(2*n, n)/(n+1) end:
A:= proc(n, k) option remember; k^n*ctln(n) end:
seq(seq(A(n, d-n), n=0..d), d=0..10);  # Alois P. Heinz, Oct 28 2019

Table[Function[k, SeriesCoefficient[2/(1 + Sqrt[1 - 4 k x]), {x, 0, n}]][j - n], {j, 0, 10}, {n, 0, j}] // Flatten
Table[Function[k, SeriesCoefficient[1/(1 + ContinuedFractionK[-k x, 1, {i, 1, n}]), {x, 0, n}]][j - n], {j, 0, 10}, {n, 0, j}] // Flatten

A(n,k) = k^n*(2*n)!/(n!*(n + 1)!).
A(n,k) = k^n*A000108(n).
G.f. of column k: 2/(1 + sqrt(1 - 4*k*x)).
G.f. of column k: 1/(1 - k*x/(1 - k*x/(1 - k*x/(1 - k*x/(1 - k*x/(1 - ...)))))), a continued fraction.
E.g.f. of column k: (BesselI(0,2*k*x) - BesselI(1,2*k*x))*exp(2*k*x).
If g.f. = 2/(1 + sqrt(1 - 4*k*x)), then a(n) ~ k^n*4^n/(sqrt(Pi)*n^(3/2)).
A(n,k) = Sum_{i=0..k} binomial(k,i) * A256061(n,k-i). - Alois P. Heinz, Oct 28 2019
For fixed k >= 1, Sum_{n>=0} 1/A(n,k) = 2*k*(8*k + 1) / (4*k - 1)^2 + 24 * k^2 * arcsin(1/(2*sqrt(k))) / (4*k - 1)^(5/2). - Vaclav Kotesovec, Nov 23 2021
For fixed k >= 1, Sum_{n>=0} (-1)^n / A(n,k) = 2*k*(8*k - 1) / (4*k + 1)^2 - 24 * k^2 * log((1 + sqrt(4*k + 1))/(2*sqrt(k))) / (4*k + 1)^(5/2). - Vaclav Kotesovec, Nov 24 2021

A256061 Number T(n,k) of 2n-length strings of balanced parentheses of exactly k different types; triangle T(n,k), n>=0, 0<=k<=n, read by rows.

Original entry on oeis.org

1, 0, 1, 0, 2, 4, 0, 5, 30, 30, 0, 14, 196, 504, 336, 0, 42, 1260, 6300, 10080, 5040, 0, 132, 8184, 71280, 205920, 237600, 95040, 0, 429, 54054, 774774, 3603600, 7207200, 6486480, 2162160, 0, 1430, 363220, 8288280, 58378320, 180180000, 273873600, 201801600, 57657600

Offset: 0

Alois P. Heinz, Mar 13 2015

Also number of binary trees with n inner nodes of exactly k different dimensions.  T(2,2) = 4:
: balanced parentheses :  ([]) :  [()] : ()[]  : []()  :
:----------------------:-------:-------:-------:-------:
:                trees :   (1) :   [2] : (1)   : [2]   :
:                      :   / \ :   / \ : / \   : / \   :
:                      : [2]   : (1)   :   [2] :   (1) :
:                      : / \   : / \   :   / \ :   / \ :

			A(3,2) = 30: (())[], (()[]), (([])), ()()[], ()([]), ()[()], ()[[]], ()[](), ()[][], ([()]), ([[]]), ([]()), ([])(), ([])[], ([][]), [(())], [()()], [()[]], [()](), [()][], [([])], [[()]], [[]()], [[]](), [](()), []()(), []()[], []([]), [][()], [][]().
Triangle T(n,k) begins:
  1;
  0,   1;
  0,   2,     4;
  0,   5,    30,     30;
  0,  14,   196,    504,     336;
  0,  42,  1260,   6300,   10080,    5040;
  0, 132,  8184,  71280,  205920,  237600,   95040;
  0, 429, 54054, 774774, 3603600, 7207200, 6486480, 2162160;
  ...

Alois P. Heinz, Rows n = 0..140, flattened

Columns k=0-1 give: A000007, A000108 (for n>0).
Main diagonal gives A001761.
Cf. A253180, A255982, A258427, A290605.

ctln:= proc(n) option remember; binomial(2*n, n)/(n+1) end:
A:= proc(n, k) option remember; k^n*ctln(n) end:
T:= (n, k)-> add(A(n, k-i)*(-1)^i*binomial(k, i), i=0..k):
seq(seq(T(n, k), k=0..n), n=0..10);

A[0, 0] = 1; A[n_, k_] := A[n, k] = k^n*CatalanNumber[n]; T[n_, k_] := Sum[A[n, k-i]*(-1)^i*Binomial[k, i], {i, 0, k}]; Table[T[n, k], {n, 0, 10}, {k, 0, n}] // Flatten (* Jean-François Alcover, Feb 20 2017, translated from Maple *)

T(n,k) = Sum_{i=0..k} (-1)^i * C(k,i) * (k-i)^n * A000108(n).
T(n,k) = k! * A253180(n,k).
T(n,k) = Sum_{i=0..k} (-1)^i * C(k,i) * A290605(n,k-i). - Alois P. Heinz, Oct 28 2019

A064299 a(n) = B(n)*C(n), where B(n) are Bell numbers (A000110) and C(n) are Catalan numbers (A000108).

Original entry on oeis.org

1, 1, 4, 25, 210, 2184, 26796, 376233, 5920200, 102816714, 1947916100, 39890416020, 876478739164, 20537052247300, 510548782729680, 13407568735200525, 370553407586717490, 10742998644116921160, 325786278993936753300, 10307990595756667951830

Offset: 0

Karol A. Penson, Sep 05 2001

nonn

From Joerg Arndt, Oct 22 2012: (Start)

Number of strings of length 2*n of up to n different types t(k) of balanced parentheses, where the first appearance of type t(k) must precede the appearance of t(k+1) for all k

For example, from the 5 parenthesis string of length 3

1: ()()(); 2: ()(()); 3: (())(); 4: (()()); 5: ((())).

we obtain the B(3) * C(3) = 5 * 5 = 25 strings

1: ()()(), ()()[], ()[](), ()[][], ()[]{};

2: ()(()), ()([]), ()[()], ()[[]], ()[{}];

3: (())(), (())[], ([])(), ([])[], ([]){};

4: (()()), (()[]), ([]()), ([][]), ([]{});

5: ((())), (([])), ([()]), ([[]]), ([{}]).

(End)

Alois P. Heinz, Table of n, a(n) for n = 0..400
K. A. Penson and J.-M. Sixdeniers, Integral Representations of Catalan and Related Numbers, J. Integer Sequences, 4 (2001), #01.2.5.

Cf. A000108, A000110.

Row sums of A253180.

with(combinat):
ctln:= proc(n) option remember; binomial(2*n, n)/(n+1) end:
a:= n-> bell(n)*ctln(n):
seq(a(n), n=0..25);  # Alois P. Heinz, Mar 23 2015

a[n_] := BellB[n]*CatalanNumber[n]; Table[a[n], {n, 0, 25}] (* Jean-François Alcover, Feb 25 2017 *)

from itertools import count, accumulate, islice
def A064299_gen(): # generator of terms
    yield from (1,1)
    blist, b, m = (1,2), 1, 1
    for n in count(1):
        blist = list(accumulate(blist, initial=(b:=blist[-1])))
        yield b*(m := m*(4*n+2)//(n+2))
A064299_list = list(islice(A064299_gen(),20)) # Chai Wah Wu, Jun 22 2022

[bell_number(i)*catalan_number(i) for i in range(17)] # Zerinvary Lajos, Mar 14 2009

Integral representation as n-th moment of a positive function on a positive half-axis, in Maple notation: a(n) = int(x^n*sum(sqrt((4*k-x)/x)*Heaviside(4*k-x)/(k*k!), k = 1..infinity)/(2*Pi*exp(1)), x = 0..infinity); this representation is unique.

A258399 Number of 4n-length strings of balanced parentheses of exactly n different types that are introduced in ascending order.

Original entry on oeis.org

1, 2, 98, 11880, 2432430, 714249900, 275335499824, 131928199603200, 75727786603836510, 50713478000403718500, 38843740303576863755100, 33508462196084294380001040, 32157574295254903735909896240, 33990046387543889224733323929120

Offset: 0

Alois P. Heinz, May 28 2015

nonn

			a(0) = 1: the empty string.
a(1) = 2: ()(), (()).
a(2) = A000108(4) * (2^3-1) = 14*7 = 98.

Alois P. Heinz, Table of n, a(n) for n = 0..250

Cf. A000108, A210029, A242446, A253180, A256254, A258426.

ctln:= proc(n) option remember; binomial(2*n, n)/(n+1) end:
A:= proc(n, k) option remember; k^n*ctln(n) end:
a:= n-> add(A(2*n, n-i)*(-1)^i/((n-i)!*i!), i=0..n):
seq(a(n), n=0..15);

A[n_, k_] := A[n, k] = k^n CatalanNumber[n];
a[n_] := If[n==0, 1, Sum[A[2n, n-i] (-1)^i/((n-i)! i!), {i, 0, n}]];
a /@ Range[0, 15] (* Jean-François Alcover, Jan 01 2021, after Alois P. Heinz *)

a(n) = A253180(2n,n).
a(n) ~ c * d^n * n! / n^(5/2), where d = A256254 = -64/(LambertW(-2*exp(-2))*(2 + LambertW(-2*exp(-2)))) = 98.8248737517356857317..., c = 1/(2^(5/2) * Pi^(3/2) * sqrt(1 + LambertW(-2*exp(-2)))) = 0.0412044746356859529237459292541572856326... . - Vaclav Kotesovec, Jun 01 2015, updated Sep 27 2023
a(n) = A210029(n) * (4*n)! / (n! * (2*n)! * (2*n + 1)!), for n>0. - Vaclav Kotesovec, Sep 27 2023

A258390 Number of 2n-length strings of balanced parentheses of exactly 2 different types that are introduced in ascending order.

Original entry on oeis.org

2, 15, 98, 630, 4092, 27027, 181610, 1239810, 8582756, 60138078, 425800564, 3042175500, 21906338040, 158830645635, 1158564772890, 8496271312650, 62604582047700, 463275674416170, 3441483002640540, 25654715940496500, 191852749820189640, 1438895966711035950

Offset: 2

Alois P. Heinz, May 28 2015

nonn

Alois P. Heinz, Table of n, a(n) for n = 2..1000

Column k=2 of A253180.

a:= proc(n) option remember; `if`(n<3, [0$2, 2][n+1],
      (2*n-1)*(6*n*a(n-1) -8*(2*n-3)*a(n-2))/(n*(n+1)))
    end:
seq(a(n), n=2..25);

Table[(2^(n-1)-1)*Binomial[2n,n]/(n+1),{n,2,20}] (* Vaclav Kotesovec, Jun 01 2015 *)

a(n) = (2*n-1)*(6*n*a(n-1)-8*(2*n-3)*a(n-2))/(n*(n+1)) for n>2, a(2)=2, a(n)=0 for n<2.

a(n) = (2^(n-1)-1) * binomial(2n,n)/(n+1) = (2^(n-1)-1)*A000108(n). - Vaclav Kotesovec, Jun 01 2015

A258391 Number of 2n-length strings of balanced parentheses of exactly 3 different types that are introduced in ascending order.

Original entry on oeis.org

5, 84, 1050, 11880, 129129, 1381380, 14707550, 156706680, 1675459786, 17998446312, 194361212500, 2110052926800, 23026236054345, 252513376831620, 2781895215981750, 30778564965687000, 341873708072702190, 3811170628172227320, 42628644369844747500

Offset: 3

Alois P. Heinz, May 28 2015

nonn

Alois P. Heinz, Table of n, a(n) for n = 3..900

Column k=3 of A253180.

Recurrence: (n-1)*n*(n+1)*a(n) = 12*(n-1)*n*(2*n - 1)*a(n-1) - 44*(n-1)*(2*n - 3)*(2*n - 1)*a(n-2) + 48*(2*n - 5)*(2*n - 3)*(2*n - 1)*a(n-3). - Vaclav Kotesovec, Jun 01 2015

a(n) ~ 12^n / (6*sqrt(Pi)*n^(3/2)). - Vaclav Kotesovec, Jun 01 2015

A258392 Number of 2n-length strings of balanced parentheses of exactly 4 different types that are introduced in ascending order.

Original entry on oeis.org

14, 420, 8580, 150150, 2432430, 37777740, 572827580, 8568059500, 127199546012, 1881416537000, 27792098497800, 410634370077750, 6074408847920670, 90017212151219100, 1336818529866319500, 19898794932394570500, 296909055625560798420, 4440849374395184751000

Offset: 4

Alois P. Heinz, May 28 2015

nonn

Alois P. Heinz, Table of n, a(n) for n = 4..800

Column k=4 of A253180.

Recurrence: (n-2)*(n-1)*n*(n+1)*a(n) = 20*(n-2)*(n-1)*n*(2*n - 1)*a(n-1) - 140*(n-2)*(n-1)*(2*n - 3)*(2*n - 1)*a(n-2) + 400*(n-2)*(2*n - 5)*(2*n - 3)*(2*n - 1)*a(n-3) - 384*(2*n - 7)*(2*n - 5)*(2*n - 3)*(2*n - 1)*a(n-4). - Vaclav Kotesovec, Jun 01 2015

a(n) ~ 16^n / (24*sqrt(Pi)*n^(3/2)). - Vaclav Kotesovec, Jun 01 2015

A258393 Number of 2n-length strings of balanced parentheses of exactly 5 different types that are introduced in ascending order.

Original entry on oeis.org

42, 1980, 60060, 1501500, 33795762, 714249900, 14504269780, 286931752800, 5578065392900, 107178276605400, 2043352620527400, 38758743724018500, 732849800716048290, 13831507110589591500, 260829110106412824900, 4917878997439418010000, 92758042341429880435020

Offset: 5

Alois P. Heinz, May 28 2015

nonn

Alois P. Heinz, Table of n, a(n) for n = 5..750

Column k=5 of A253180.

Recurrence: (n-3)*(n-2)*(n-1)*n*(n+1)*a(n) = 30*(n-3)*(n-2)*(n-1)*n*(2*n - 1)*a(n-1) - 340*(n-3)*(n-2)*(n-1)*(2*n - 3)*(2*n - 1)*a(n-2) + 1800*(n-3)*(n-2)*(2*n - 5)*(2*n - 3)*(2*n - 1)*a(n-3) - 4384*(n-3)*(2*n - 7)*(2*n - 5)*(2*n - 3)*(2*n - 1)*a(n-4) + 3840*(2*n - 9)*(2*n - 7)*(2*n - 5)*(2*n - 3)*(2*n - 1)*a(n-5). - Vaclav Kotesovec, Jun 01 2015

a(n) ~ 20^n / (120*sqrt(Pi)*n^(3/2)). - Vaclav Kotesovec, Jun 01 2015

A258394 Number of 2n-length strings of balanced parentheses of exactly 6 different types that are introduced in ascending order.

Original entry on oeis.org

132, 9009, 380380, 12864852, 383402292, 10551322782, 275335499824, 6924802684800, 169656773406120, 4078556074277685, 96700630711999860, 2269529269318731420, 52868514692841609300, 1224857602490265215010, 28265620407321158141280, 650452332645092821924080

Offset: 6

Alois P. Heinz, May 28 2015

nonn

Alois P. Heinz, Table of n, a(n) for n = 6..700

Column k=6 of A253180.

Recurrence: (n-4)*(n-3)*(n-2)*(n-1)*n*(n+1)*a(n) = 42*(n-4)*(n-3)*(n-2)*(n-1)*n*(2*n - 1)*a(n-1) - 700*(n-4)*(n-3)*(n-2)*(n-1)*(2*n - 3)*(2*n - 1)*a(n-2) + 5880*(n-4)*(n-3)*(n-2)*(2*n - 5)*(2*n - 3)*(2*n - 1)*a(n-3) - 25984*(n-4)*(n-3)*(2*n - 7)*(2*n - 5)*(2*n - 3)*(2*n - 1)*a(n-4) + 56448*(n-4)*(2*n - 9)*(2*n - 7)*(2*n - 5)*(2*n - 3)*(2*n - 1)*a(n-5) - 46080*(2*n - 11)*(2*n - 9)*(2*n - 7)*(2*n - 5)*(2*n - 3)*(2*n - 1)*a(n-6). - Vaclav Kotesovec, Jun 01 2015

a(n) ~ 24^n / (720*sqrt(Pi)*n^(3/2)). - Vaclav Kotesovec, Jun 01 2015

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A002740 Number of tree-rooted bridgeless planar maps with two vertices and n faces.

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A290605 Square array A(n,k), n>=0, k>=0, read by antidiagonals, where column k is the expansion of 2/(1 + sqrt(1 - 4*k*x)).

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A256061 Number T(n,k) of 2n-length strings of balanced parentheses of exactly k different types; triangle T(n,k), n>=0, 0<=k<=n, read by rows.

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A064299 a(n) = B(n)*C(n), where B(n) are Bell numbers (A000110) and C(n) are Catalan numbers (A000108).

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A258399 Number of 4n-length strings of balanced parentheses of exactly n different types that are introduced in ascending order.

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A258390 Number of 2n-length strings of balanced parentheses of exactly 2 different types that are introduced in ascending order.

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A258391 Number of 2n-length strings of balanced parentheses of exactly 3 different types that are introduced in ascending order.

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A290605 Square array A(n,k), n>=0, k>=0, read by antidiagonals, where column k is the expansion of 2/(1 + sqrt(1 - 4kx)).