A057977 -id:A057977 - cp's OEIS frontend

A080385 Numbers k such that there are exactly 7 numbers j for which binomial(k, floor(k/2)) / binomial(k,j) is an integer, i.e., A080383(k) = 7.

12, 30, 56, 84, 90, 132, 154, 182, 220, 252, 280, 306, 312, 340, 374, 380, 408, 418, 440, 456, 462, 476, 532, 552, 598, 616, 624, 630, 644, 650, 660, 690, 756, 828, 840, 858, 870, 880, 884, 900, 918, 936, 952, 966, 986, 992, 1020, 1054, 1102, 1116, 1140, 1160

Offset: 1

Labos Elemer, Mar 12 2003

nonn

			For n=12, the central binomial coefficient (C(12,6) = 924) is divisible by C(12,0), C(12,1), C(12,2), C(12,6), C(12,10), C(12,11), and C(12,12).

Vaclav Kotesovec, Table of n, a(n) for n = 1..4963

Cf. A327430, A080384, A080386, A327431, A080387.

Cf. A001405, A057977.

More terms from Vaclav Kotesovec, Sep 06 2019

A238762 Triangle read by rows, generalized ballot numbers, 0<=k<=n.

Original entry on oeis.org

1, 0, 1, 1, 0, 1, 0, 2, 0, 3, 1, 0, 2, 0, 2, 0, 3, 0, 8, 0, 10, 1, 0, 3, 0, 5, 0, 5, 0, 4, 0, 15, 0, 30, 0, 35, 1, 0, 4, 0, 9, 0, 14, 0, 14, 0, 5, 0, 24, 0, 63, 0, 112, 0, 126, 1, 0, 5, 0, 14, 0, 28, 0, 42, 0, 42, 0, 6, 0, 35, 0, 112, 0, 252, 0, 420, 0, 462

Offset: 0

Peter Luschny, Mar 05 2014

Compare with the definition of the Motzkin triangle A238763.

			[n\k 0  1  2   3  4   5  6   7]
[0]  1,
[1]  0, 1,
[2]  1, 0, 1,
[3]  0, 2, 0,  3,
[4]  1, 0, 2,  0, 2,
[5]  0, 3, 0,  8, 0, 10,
[6]  1, 0, 3,  0, 5,  0, 5,
[7]  0, 4, 0, 15, 0, 30, 0, 35.

D. E. Knuth, TAOCP, Vol. 4a, Section 7.2.1.6, Eq. 22, p. 451.

P. Luschny, The lost Catalan numbers

Cf. A009766, A189231, A238761, A238879.

binom2 := proc(n, k) local h;
   h := n -> (n-((1-(-1)^n)/2))/2;
   n!/(h(n-k)!*h(n+k)!) end:
A238762 := proc(n, k) local a,b,c;
   a := iquo(n+k+2+modp(n,2), 2);
   b := iquo(n-k+2, 2);
   c := modp(n+k+1, 2);
   binom2(a,b)*b*c/a end:
seq(print(seq(A238762(n, k), k=0..n)), n=0..7);
# Alternativ:
ballot := proc(p, q) option remember;
    if p = 0 and q = 0 then return 1 fi;
    if p < 0 or  p > q then return 0 fi;
    ballot(p-2, q) + ballot(p, q-2);
    if type(q, odd) then % + ballot(p-1, q-1) fi;
    % end:

T[n_, k_] := T[n, k] = Which[k == 0 && n == 0, 1, k < 0 || k > n, 0, True, s = T[n, k - 2] + T[n - 2, k]; If[OddQ[n], s += T[n - 1, k - 1]]; s];
Table[T[n, k], {n, 0, 11}, {k, 0, n}] // Flatten (* Jean-François Alcover, Jul 10 2019, adapted from Sage code *)

@CachedFunction
def ballot(p, q):
    if p == 0 and q == 0: return 1
    if p < 0 or p > q: return 0
    S = ballot(p-2, q) + ballot(p, q-2)
    if q % 2 == 1: S += ballot(p-1, q-1)
    return S
for q in range(8): [ballot(p, q) for p in (0..q)]

Definition: T(0, 0) = 1; T(p, q) = 0 if p < 0 or p > q; T(p, q) = T(p-2, q) + (q mod 2) T(p-1, q-1) + T(p, q-2). (The notation is in the style of Knuth, TAOCP 4a (7.2.1.6)).
T(2*k, 2*n) are the classical ballot numbers A009766(n, k).
T(2*k-1, 2*n-1) = A238761(n, k).
T(n,k) = c*A189231(a, b) with a = floor((n + k + (k mod 2))/2), b = floor((n-k)/2) and c = ((n+k+1) mod 2).
T(n, k) = ((n+k+1) mod 2)*((floor(n/2)+floor(k/2) + 1)^(k mod 2)) * (binomial(floor(n/2) + floor(k/2), floor(n/2)) - binomial(floor(n/2) + floor(k/2), floor(n/2) + 1)).
T(n, k) = ((n+k+1) mod 2)*((floor(n/2)+floor(k/2) + 1)^(k mod 2)) * (floor((n-k)/2) + 1)/(floor(n/2) + 1) * binomial(floor(n/2) + floor(k/2), floor(n/2)).
T(n, n) = A057977(n).
T(n, n-2) = A238452(n-1).
Row sums are A238879.

A080384 Numbers k such that there are exactly 6 numbers j for which binomial(k, floor(k/2)) / binomial(k,j) is an integer, i.e., A080383(k) = 6.

Original entry on oeis.org

5, 7, 9, 11, 15, 17, 19, 21, 23, 27, 29, 33, 35, 39, 43, 45, 47, 49, 51, 53, 55, 59, 61, 63, 65, 67, 69, 71, 73, 75, 77, 79, 81, 83, 87, 89, 93, 95, 97, 99, 101, 103, 105, 107, 109, 111, 113, 115, 117, 119, 121, 123, 125, 127, 129, 131, 135, 137, 139, 141, 143, 145

Offset: 1

Labos Elemer, Mar 12 2003

nonn

			For n=9, the central binomial coefficient (C(9,4) = 126) is divisible by C(9,0), C(9,1), C(9,4), C(9,5), C(9,8), and C(9,9); certain primes are missing, certain composites are here.

Vaclav Kotesovec, Table of n, a(n) for n = 1..44084

Cf. A327430, A080385, A080386, A327431, A080387.

Cf. A001405, A057977.

Position[Table[Count[Binomial[n,Floor[n/2]]/Binomial[n,Range[0,n]],?IntegerQ],{n,150}],6]//Flatten (* _Harvey P. Dale, Mar 05 2023 *)

isok(n) = my(b=binomial(n, n\2)); sum(i=0, n, (b % binomial(n, i)) == 0) == 6; \\ Michel Marcus, Jul 29 2017

A080386 Numbers k such that there are exactly 8 numbers j for which binomial(k, floor(k/2)) / binomial(k,j) is an integer, i.e., A080383(k) = 8.

Original entry on oeis.org

25, 37, 169, 199, 201, 241, 397, 433, 547, 685, 865, 1045, 1081, 1585, 1657, 1891, 1951, 1969, 2071, 2143, 2647, 2901, 3011, 3025, 3097, 3151, 3251, 3421, 3511, 3727, 4105, 4213, 4453, 4771, 4885, 5581, 5857, 6019, 6031, 6265, 6397, 6967, 7345, 7615, 7831, 8425, 8857, 8929

Offset: 1

Labos Elemer, Mar 12 2003

nonn

			For n=25, the central binomial coefficient (C(25,12) = 5200300) is divisible by C(25,0), C(25,1), C(25,3), C(25,12), C(25,13), C(25,22), C(25,24), and C(25,25).

Vaclav Kotesovec, Table of n, a(n) for n = 1..260

Cf. A327430, A080384, A080385, A327431, A080387.

Cf. A001405, A057977.

More terms from Michel Marcus, Aug 23 2019

A080387 Numbers k such that there are exactly 10 numbers j for which binomial(k, floor(k/2)) / binomial(k,j) is an integer, i.e., A080383(k) = 10.

Original entry on oeis.org

13, 31, 41, 57, 85, 91, 133, 155, 177, 183, 209, 221, 253, 281, 307, 313, 341, 375, 381, 409, 419, 441, 457, 463, 477, 481, 533, 553, 599, 617, 625, 631, 645, 651, 661, 691, 737, 757, 829, 841, 859, 871, 881, 885, 901, 919, 929, 937, 953, 967, 987, 993

Offset: 1

Labos Elemer, Mar 12 2003

nonn

			For n=13, the central binomial coefficient (C(13,6) = 1716) is divisible by 10 binomial coefficients C(13,j); the 4 nondivisible cases are C(13,4), C(13,5), C(13,8), and C(13,9).

Vaclav Kotesovec, Table of n, a(n) for n = 1..5113

Cf. A327430, A080384, A080385, A080386, A327431.

Cf. A001405, A057977.

A189231 Extended Catalan triangle read by rows.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 3, 2, 3, 1, 2, 8, 3, 4, 1, 10, 5, 15, 4, 5, 1, 5, 30, 9, 24, 5, 6, 1, 35, 14, 63, 14, 35, 6, 7, 1, 14, 112, 28, 112, 20, 48, 7, 8, 1, 126, 42, 252, 48, 180, 27, 63, 8, 9, 1, 42, 420, 90, 480, 75, 270, 35, 80, 9, 10, 1, 462, 132, 990, 165, 825, 110, 385, 44, 99, 10, 11, 1

Offset: 0

Peter Luschny, May 01 2011

Let S(n,k) denote the coefficients of the positive powers of the Laurent polynomials C_n(x) = (x+1/x)^(n-1)*(x-1/x)*(x+1/x+n) (if n>0) and C_0(x) = 0.
Then T(n,k) = S(n+1,k+1) for n>=0, k>=0.
The classical Catalan triangle A053121(n,k) can be recovered from this triangle by setting T(n,k) = 0 if n-k is odd.
The complementary Catalan triangle A189230(n,k) can be recovered from this triangle by setting T(n,k) = 0 if n-k is even.
T(n,0) are the extended Catalan numbers A057977(n).

			The Laurent polynomials:
C(0,x) =                 0
C(1,x) =               x - 1/x
C(2,x) =         x^2 + x - 1/x - 1/x^2
C(3,x) = x^3 + 2 x^2 + x - 1/x - 2/x^2 -1/x^3
Triangle T(n,k) = S(n+1,k+1) starts
[0]   1,
[1]   1,  1,
[2]   1,  2,  1,
[3]   3,  2,  3,  1,
[4]   2,  8,  3,  4,  1,
[5]  10,  5, 15,  4,  5,  1,
[6]   5, 30,  9, 24,  5,  6,  1,
[7]  35, 14, 63, 14, 35,  6,  7, 1,
    [0],[1],[2],[3],[4],[5],[6],[7]

Peter Luschny, Die schwingende Fakultät und Orbitalsysteme, August 2011.
Peter Luschny, The lost Catalan numbers

Cf. A053121, A162246, A057977, A189230.

A189231_poly := (n,x)-> `if`(n=0,0,(x+1/x)^(n-2)*(x-1/x)*(x+1/x+n-1)):
seq(print([n],seq(coeff(expand(A189231_poly(n,x)),x,k),k=1..n)),n=1..9);
A189231 := proc(n,k) option remember; `if`(k>n or k<0, 0, `if`(n=k, 1, A189231(n-1,k-1)+modp(n-k,2)*A189231(n-1,k)+A189231(n-1,k+1))) end:
seq(print(seq(A189231(n,k),k=0..n)),n=0..9);

t[n_, k_] /; (k > n || k < 0) = 0; t[n_, n_] = 1; t[n_, k_] := t[n, k] = t[n-1, k-1] + Mod[n-k, 2]*t[n-1, k] + t[n-1, k+1]; Table[t[n, k], {n, 0, 11}, {k, 0, n}] // Flatten (* Jean-François Alcover, Jul 30 2013 *)

Recurrence: If k>n or k<0 then T(n,k) = 0 else if n=k then T(n,k) = 1; otherwise T(n,k) = T(n-1,k-1) + ((n-k) mod 2)*T(n-1,k) + T(n-1,k+1).
S(n,k) = (k/n)* A162246(n,k) for n>0 where S(n,k) are the coefficients from the definition provided the triangle A162246 is indexed in Laurent style by the recurrence: if abs(k) > n then A162246(n,k) = 0 else if n = k then A162246(n,k) = 1 and otherwise A162246(n,k) = A162246(n-1,k-1)+ modp(n-k,2) * A162246(n-1,k) + A162246(n-1,k+1).
Row sums: A189911(n) = A162246(n,n) + A162246(n,n+1) for n>0.

A063549 Smallest number of crossing-free matchings on n points in the plane.

Original entry on oeis.org

1, 1, 3, 2, 10, 5, 35, 14, 126, 42, 462, 132, 1716, 429, 6435, 1430, 24310, 4862, 92378, 16796, 352716, 58786, 1352078, 208012, 5200300, 742900, 20058300, 2674440, 77558760, 9694845, 300540195, 35357670, 1166803110, 129644790, 4537567650

Offset: 1

N. J. A. Sloane, Aug 14 2001

a(n) = Catalan(n/2) if n is even else n*Catalan((n-1)/2) (see Garcia reference). The same as A057977. - Vladeta Jovovic, Mar 20 2010

O. Aichholzer and H. Krasser, The point set order type data base: a collection of applications and results, pp. 17-20 in Abstracts 13th Canadian Conference on Computational Geometry (CCCG '01), Waterloo, Aug. 13-15, 2001.
A. Garcia, M. Noy, and J. Tejel, Lower bounds on the number of crossing-free subgraphs of K_N, Comput. Geom., 16 (2000), pp. 211-221.

Cf. A057977, A063550.

# See A057977 for an implementation based on ballot numbers. Peter Luschny, Feb 23 2019

a[n_?EvenQ] := CatalanNumber[n/2]; a[n_?OddQ] := n*CatalanNumber[(n-1)/2]; Table[a[n], {n, 3, 35}] (* Jean-François Alcover, Feb 03 2012, after Vladeta Jovovic *)

(n+2)*a(n) -n*a(n-1) +4*(-2*n+1)*a(n-2) +4*(n-1)*a(n-3) +16*(n-3)*a(n-4)=0. - R. J. Mathar, Jun 13 2013

Sum_{n>=1} 1/a(n) = 5/3 + 8*Pi/(9*sqrt(3)). - Amiram Eldar, Aug 20 2022

More terms from Jean-François Alcover, Feb 03 2012

a(1) = a(2) = 1 inserted and added Garcia reference from Nathaniel Johnston, Nov 17 2014

A224747 Number of lattice paths from (0,0) to (n,0) that do not go below the x-axis and consist of steps U=(1,1), D=(1,-1) and H=(1,0), where H-steps are only allowed if y=1.

Original entry on oeis.org

1, 0, 1, 1, 3, 5, 12, 23, 52, 105, 232, 480, 1049, 2199, 4777, 10092, 21845, 46377, 100159, 213328, 460023, 981976, 2115350, 4522529, 9735205, 20836827, 44829766, 96030613, 206526972, 442675064, 951759621, 2040962281, 4387156587, 9411145925, 20226421380

Offset: 0

Alois P. Heinz, Apr 17 2013

nonn

Also the number of non-capturing (cf. A054391) set-partitions of {1..n} without singletons. - Christian Sievers, Oct 29 2024

			a(5) = 5: UHHHD, UDUHD, UUDHD, UHDUD, UHUDD.
a(6) = 12: UHHHHD, UDUHHD, UUDHHD, UHDUHD, UHUDHD, UHHDUD, UDUDUD, UUDDUD, UHHUDD, UDUUDD, UUDUDD, UUUDDD.
G.f. = 1 + x^2 + x^3 + 3*x^4 + 5*x^5 + 12*x^6 + 23*x^7 + 52*x^8 + 105*x^9 + ...

Alois P. Heinz, Table of n, a(n) for n = 0..1000
C. Banderier and M. Wallner, Lattice paths with catastrophes, SLC 77, Strobl - 12.09.2016, H(x).
Cyril Banderier and Michael Wallner, Lattice paths with catastrophes, arXiv:1707.01931 [math.CO], 2017.
Jean-Luc Baril and Sergey Kirgizov, Bijections from Dyck and Motzkin meanders with catastrophes to pattern avoiding Dyck paths, arXiv:2104.01186 [math.CO], 2021.
Jean-Luc Baril, Sergey Kirgizov, and Armen Petrossian, Dyck Paths with catastrophes modulo the positions of a given pattern, Australasian J. Comb. (2022) Vol. 84, No. 2, 398-418.

Cf. A000108 (without H-steps), A001006 (unrestricted H-steps), A057977 (<=1 H-step).
Cf. A000012, A101455, A125187, A001405 (invert transform).
Inverse binomial transform of A054391.

a:= proc(n) option remember; `if`(n<5, [1, 0, 1, 1, 3][n+1],
      a(n-1)+ (6*(n-3)*a(n-2) -3*(n-5)*a(n-3)
      -8*(n-4)*a(n-4) -4*(n-4)*a(n-5))/(n-1))
    end:
seq(a(n), n=0..40);

a[n_] := a[n] = If[n < 5, {1, 0, 1, 1, 3}[[n+1]], a[n-1] + (6*(n-3)*a[n-2] - 3*(n-5)*a[n-3] - 8*(n-4)*a[n-4] - 4*(n-4)*a[n-5])/(n-1)]; Table[a[n], {n, 0, 34}] (* Jean-François Alcover, Jun 20 2013, translated from Maple *)
a[ n_] := SeriesCoefficient[ (2 - 3 x - 2 x^2 + x Sqrt[1 - 4 x^2]) / (2 (1 - x - 2 x^2 - x^3)), {x, 0, n}] (* Michael Somos, Jan 14 2014 *)

{a(n) = if( n<0, 0, polcoeff( (2 - 3*x - 2*x^2 + x * sqrt(1 - 4*x^2 + x * O(x^n)) ) / (2 * (1 - x - 2*x^2 - x^3)), n))} /* Michael Somos, Jan 14 2014 */

a(n) = Sum_{k=0..floor((n-2)/2)} A009766(2*n-3*k-3, k) for n >= 2. - Johannes W. Meijer, Jul 22 2013
a(2*n) = A125187(n) (bisection). - R. J. Mathar, Jul 27 2013
HANKEL transform is A000012. HANKEL transform omitting a(0) is a period 4 sequence [0, -1, 0, 1, ...] = -A101455. - Michael Somos, Jan 14 2014
Given g.f. A(x), then 0 = A(x)^2 * (x^3 + 2*x^2 + x - 1) + A(x) * (-2*x^2 - 3*x + 2) + (2*x - 1). - Michael Somos, Jan 14 2014
0 = a(n)*(a(n+1) +2*a(n+2) +a(n+3) -a(n+4)) +a(n+1)*(2*a(n+1) +5*a(n+2) +a(n+3) -2*a(n+4)) +a(n+2)*(2*a(n+2) -a(n+3) -a(n+4)) +a(n+3)*(-a(n+3) +a(n+4)). - Michael Somos, Jan 14 2014
G.f.: (2 - 3*x - 2*x^2 + x * sqrt(1 - 4*x^2)) / (2 * (1 - x - 2*x^2 - x^3)). - Michael Somos, Jan 14 2014
D-finite with recurrence (-n+1)*a(n) +(n-1)*a(n-1) +6*(n-3)*a(n-2) +3*(-n+5)*a(n-3) +8*(-n+4)*a(n-4) +4*(-n+4)*a(n-5)=0. - R. J. Mathar, Sep 15 2021

A077587 a(n) = C(n+1) + n*C(n) where C = A000108 (Catalan numbers).

Original entry on oeis.org

1, 3, 9, 29, 98, 342, 1221, 4433, 16302, 60554, 226746, 854658, 3239044, 12332140, 47137005, 180780345, 695367510, 2681600130, 10364759790, 40142121030, 155748675420, 605274171060, 2355676013730, 9180275261274, 35819645937228

Offset: 0

Michael Somos, Nov 09 2002

Number of ascents of length 2 starting at an even level in all Dyck paths of semilength n+2. Example: a(1)=3 because all Dyck paths of semilength 3 are UDUDUD, UD(UU)DD, (UU)DDUD, (UU)DUDD and UUUDDD, where U=(1,1), D=(1,-1), having altogether 3 ascents of length 2 that start at an even level (shown between parentheses). - Emeric Deutsch, Nov 29 2005

a(n) is the number of parking functions of size n+1 avoiding the patterns 132, 231, and 321. - Lara Pudwell, Apr 10 2023

G. C. Greubel and Vincenzo Librandi, Table of n, a(n) for n = 0..1000(terms 0..200 from Vincenzo Librandi)
Ayomikun Adeniran and Lara Pudwell, Pattern avoidance in parking functions, Enumer. Comb. Appl. 3:3 (2023), Article S2R17.
A. Asinowski and G. Rote, Point sets with many non-crossing matchings, arXiv:1502.04925 [cs.CG], 2015.

Cf. A000108, A057977, A114462.

egf := x -> exp(2*x)*(1+1/x)*BesselI(1, 2*x);
seq(n!*coeff(series(egf(x), x, n+2), x, n), n=0..24); # Peter Luschny, Apr 14 2014

Table[(CatalanNumber[n + 1] + n CatalanNumber[n]), {n, 0, 40}] (* Vincenzo Librandi, Apr 15 2014 *)

a(n)=if(n<0,0,(n^2+6*n+2)*(2*n)!/n!/(n+2)!)

a(n)=if(n<0,0,polcoeff((4+x+1/x-(x+1/x)^2)*(1+x)^(2*n),n)/2)

a(n) = binomial(2n+1, n+1) - binomial(2n, n+2).
a(n) = (3*(3*n+2)*a(n-1) - 2*(11*n-7)*a(n-2) + 4*(2*n-5)*a(n-3))/(n+2), n>2.
G.f.: A(x) = (1 - 3*x - (1-5*x+2*x^2)/sqrt(1-4*x) )/(2*x^2) satisfies 0 = (x^2+4*x-1) + (12*x^2-7*x+1)*A + (4*x^3-x^2)*A^2.
E.g.f.: A(x) = (1+x)B(x)' where B(x) = e.g.f. of A000108.
a(n) = Sum_{k=0..n} binomial(n,k)*A057977(k)*2^(n-k); here the A057977 are understood as the extended Catalan numbers (see also A063549). Related to Touchard's identity. - Peter Luschny, Jul 14 2016
a(n) ~ 4^n/sqrt(Pi*n). - Ilya Gutkovskiy, Jul 14 2016
Asymptotic starts a(n) ~ (4^n/sqrt(Pi*n))*(1 + (23/2^3)/n - (1199/2^7)/n^2 +(22685/2^10)/n^3 - (1562421/2^15)/n^4 + ... ). - Peter Luschny, Jul 14 2016

A241543 a(n) = A241477(n, n).

Original entry on oeis.org

1, 1, 2, 2, 2, 6, 4, 20, 10, 70, 28, 252, 84, 924, 264, 3432, 858, 12870, 2860, 48620, 9724, 184756, 33592, 705432, 117572, 2704156, 416024, 10400600, 1485800, 40116600, 5348880, 155117520, 19389690, 601080390, 70715340, 2333606220, 259289580, 9075135300

Offset: 0

Peter Luschny, Apr 25 2014

nonn

See A241477 and A232500 for the combinatorial definitions.

Cf. A057977, A241477, A232500.

A241543 := proc(n)
    if n < 2 then 1
  else 2*iquo(n,2)*(n-2)!/iquo(n,2)!^2
    fi end:
seq(A241543(n), n=0..37);

a(n) = 2*floor(n/2)*(n-2)!/floor(n/2)!^2 for n>=2.

a(n+2) = 2*A057977(n) for n>=0. - Peter Luschny, Jul 17 2016

cp's OEIS Frontend

A080385 Numbers k such that there are exactly 7 numbers j for which binomial(k, floor(k/2)) / binomial(k,j) is an integer, i.e., A080383(k) = 7.

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A238762 Triangle read by rows, generalized ballot numbers, 0<=k<=n.

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A080384 Numbers k such that there are exactly 6 numbers j for which binomial(k, floor(k/2)) / binomial(k,j) is an integer, i.e., A080383(k) = 6.

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A080386 Numbers k such that there are exactly 8 numbers j for which binomial(k, floor(k/2)) / binomial(k,j) is an integer, i.e., A080383(k) = 8.

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A080387 Numbers k such that there are exactly 10 numbers j for which binomial(k, floor(k/2)) / binomial(k,j) is an integer, i.e., A080383(k) = 10.

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A189231 Extended Catalan triangle read by rows.

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Maple

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A063549 Smallest number of crossing-free matchings on n points in the plane.

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Maple

Mathematica

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A224747 Number of lattice paths from (0,0) to (n,0) that do not go below the x-axis and consist of steps U=(1,1), D=(1,-1) and H=(1,0), where H-steps are only allowed if y=1.

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