A063549 -id:A063549 - cp's OEIS frontend

A057977 GCD of consecutive central binomial coefficients: a(n) = gcd(A001405(n+1), A001405(n)).

1, 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: 0

Labos Elemer, Nov 13 2000

The numbers can be seen as a generalization of the Catalan numbers, extending A000984(n)/(n+1) to A056040(n)/(floor(n/2)+1). They can also be seen as composing the aerated Catalan numbers A126120 with the aerated complementary Catalan numbers A138364. (Thus the name 'extended Catalan numbers' might be apt for this sequence.) - Peter Luschny, May 03 2011
a(n) is the 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 maximally one step H=(1,0). - Alois P. Heinz, Apr 17 2013
Equal to A063549 (see comments in that sequence). - Nathaniel Johnston, Nov 17 2014
a(n) can be computed with ballot numbers without multiplications or divisions, see Maple program. - Peter Luschny, Feb 23 2019

			This GCD equals A001405(n) for the smaller odd number gcd(C(12,6), C(11,5)) = gcd(924,462) = 462 = C(11,5).

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Peter Luschny, Die schwingende Fakultät und Orbitalsysteme, August 2011.
Peter Luschny, The lost Catalan numbers.

Cf. A001405, A063549.

Bisections are A000108 and A001700.

A057977_ogf := proc(z) b := z -> (z-1)/(2*z^2);
(2 + b(z))/sqrt(1-4*z^2) - b(z) end:
seq(coeff(series(A057977_ogf(z),z,n+3),z,n), n = 0..35);
A057977_rec := n -> `if`(n=0, 1, A057977_rec(n-1)*n^modp(n,2)
*(4/(n+2))^modp(n+1,2));
A057977_int := proc(n) int((x^(2*n-1)*((4-x)^2/x)^cos(Pi*n))^(1/4),x=0..4)/(2*Pi); round(evalf(%)) end:
A057977 := n -> (n!/iquo(n,2)!^2) / (iquo(n,2)+1):
seq(A057977(n), n=0..35); # Peter Luschny, Apr 30 2011
b := proc(p, q) option remember; local S;
   if p = 0 and q = 0 then return 1 fi;
   if p < 0 or  p > q then return 0 fi;
   S := b(p-2, q) + b(p, q-2);
   if type(q, odd) then S := S + b(p-1, q-1) fi;
   S end:
seq(b(n, n), n=0..35); # Peter Luschny, Feb 23 2019

a[n_] := n! / (Quotient[n, 2]!^2 * (Quotient[n, 2]+1)); Table[a[n], {n, 0, 35}] (* Jean-François Alcover, Feb 03 2012, after Peter Luschny *)

a(n)=if(n<0,0,(n+n%2)!/(n\2+1)!/(n\2+n%2)!/(1+n%2))
a(n)=n!/(n\2)!^2/(n\2+1) \\ Charles R Greathouse IV, May 02 2011

def A057977():
    x, n = 1, 1
    while True:
        yield x
        m = n if is_odd(n) else 4/(n+2)
        x *= m
        n += 1
a = A057977(); [next(a) for i in range(36)]   # Peter Luschny, Oct 21 2013

G.f.: (4*x^2+x-1+(1-x)*sqrt(1-4*x^2))/(2*sqrt(1-4*x^2)*x^2). E.g.f.: (1+1/x)*BesselI(1, 2*x). - Vladeta Jovovic, Jan 19 2004
From Peter Luschny, Apr 30 2011: (Start)
Recurrence: a(0) = 1 and a(n) = a(n-1)*n^[n odd]*(4/(n+2))^[n even] for n > 0.
Asymptotic formula: Let [n even] = 1 if n is even, 0 otherwise. Let N := n+1+[n even]. Then a(n) ~ 2^N /((n+1)^[n even]*sqrt(Pi*(2*N+1))).
Integral representation: a(n) = (1/(2*Pi))*Int_{x=0..4}(x^(2*n-1)* ((4-x)^2/x)^cos(Pi*n))^(1/4) (End)
E.g.f.: U(0) where U(k)= 1 + x/(1 - x/(x + (k+1)*(k+2)//U(k+1))); (continued fraction, 3-step). - Sergei N. Gladkovskii, Oct 19 2012
From R. J. Mathar, Sep 16 2016: (Start)
D-finite with recurrence: (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.
D-finite with recurrence: -(n+2)*(n^2-5)*a(n) + 4*(-2*n-1)*a(n-1) + 4*(n-1)*(n^2+2*n-4)*a(n-2) = 0. (End)
Sum_{n>=0} 1/a(n) = 8/3 + 8*Pi/(9*sqrt(3)). - Amiram Eldar, Aug 20 2022

A274709 A statistic on orbital systems over n sectors: the number of orbitals which rise to maximum height k over the central circle.

Original entry on oeis.org

1, 1, 1, 1, 3, 3, 2, 3, 1, 10, 15, 5, 5, 9, 5, 1, 35, 63, 35, 7, 14, 28, 20, 7, 1, 126, 252, 180, 63, 9, 42, 90, 75, 35, 9, 1, 462, 990, 825, 385, 99, 11, 132, 297, 275, 154, 54, 11, 1, 1716, 3861, 3575, 2002, 702, 143, 13, 429, 1001, 1001, 637, 273, 77, 13, 1

Offset: 0

Peter Luschny, Jul 09 2016

The definition of an orbital system is given in A232500 (see also the illustration there). The number of orbitals over n sectors is counted by the swinging factorial A056040.
Note that (sum row_n) / row_n(0) = 1,1,2,2,3,3,4,4,..., i.e. the swinging factorials are multiples of the extended Catalan numbers A057977 generalizing the fact that the central binomials are multiples of the Catalan numbers.
T(n, k) is a subtriangle of the extended Catalan triangle A189231.

			Triangle read by rows, n>=0. The length of row n is floor((n+2)/2).
[ n] [k=0,1,2,...] [row sum]
[ 0] [  1] 1
[ 1] [  1] 1
[ 2] [  1,   1] 2
[ 3] [  3,   3] 6
[ 4] [  2,   3,   1] 6
[ 5] [ 10,  15,   5] 30
[ 6] [  5,   9,   5,   1] 20
[ 7] [ 35,  63,  35,   7] 140
[ 8] [ 14,  28,  20,   7,  1] 70
[ 9] [126, 252, 180,  63,  9] 630
[10] [ 42,  90,  75,  35,  9,  1] 252
[11] [462, 990, 825, 385, 99, 11] 2772
[12] [132, 297, 275, 154, 54, 11, 1] 924
T(6, 2) = 5 because the five orbitals [-1, 1, 1, 1, -1, -1], [1, -1, 1, 1, -1, -1], [1, 1, -1, -1, -1, 1], [1, 1, -1, -1, 1, -1], [1, 1, -1, 1, -1, -1] raise to maximal height of 2 over the central circle.

Peter Luschny, The lost Catalan numbers
Peter Luschny, Orbitals

Cf. A008313, A039599 (even rows), A047072, A056040 (row sums), A057977 (col 0), A063549 (col 0), A112467, A120730, A189230 (odd rows aerated), A189231, A232500.

Other orbital statistics: A241477 (first zero crossing), A274706 (absolute integral), A274708 (number of peaks), A274710 (number of turns), A274878 (span), A274879 (returns), A274880 (restarts), A274881 (ascent).

S := proc(n,k) option remember; `if`(k>n or k<0, 0, `if`(n=k, 1, S(n-1,k-1)+
modp(n-k,2)*S(n-1,k)+S(n-1,k+1))) end: T := (n,k) -> S(n,2*k);
seq(print(seq(T(n,k), k=0..iquo(n,2))), n=0..12);

from itertools import accumulate
# Brute force counting
def unit_orbitals(n):
    sym_range = [i for i in range(-n+1, n, 2)]
    for c in Combinations(sym_range, n):
        P = Permutations([sgn(v) for v in c])
        for p in P: yield p
def max_orbitals(n):
    if n == 0: return [1]
    S = [0]*((n+2)//2)
    for u in unit_orbitals(n):
        L = list(accumulate(u))
        S[max(L)] += 1
    return S
for n in (0..10): print(max_orbitals(n))

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

A063550 Largest number of crossing-free matchings on a set S of n points in the plane, that is, a set of floor(n/2) pairwise non-intersecting segments with endpoints in S having no endpoint in common.

Original entry on oeis.org

1, 1, 3, 3, 14, 12, 79, 56, 497, 311

Offset: 1

N. J. A. Sloane, Aug 14 2001

			The Sharir link contains an image (Figure 1) of a placement of 6 points in the plane such that 12 of their perfect matchings are crossing-free, demonstrating that a(6) >= 12. - _Nathaniel Johnston_, Nov 17 2014

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.
M. Sharir and E. Welzl. On the Number of Crossing-Free Matchings, (Cycles, and Partitions)

Cf. A063549.

a(1) = a(2) = 1 inserted by Nathaniel Johnston, Nov 17 2014

Name clarified by Manfred Scheucher, Mar 12 2018

A359648 Triangle read by rows. T(n, k) = (n!)^2 / (k! * (n - k)! * (floor(n/2)!)^2 * (floor(n/2) + 1)).

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 3, 9, 9, 3, 2, 8, 12, 8, 2, 10, 50, 100, 100, 50, 10, 5, 30, 75, 100, 75, 30, 5, 35, 245, 735, 1225, 1225, 735, 245, 35, 14, 112, 392, 784, 980, 784, 392, 112, 14, 126, 1134, 4536, 10584, 15876, 15876, 10584, 4536, 1134, 126

Offset: 0

Peter Luschny, Jan 09 2023

			Triangle T(n, k) starts:
[0]   1;
[1]   1,    1;
[2]   1,    2,    1;
[3]   3,    9,    9,     3;
[4]   2,    8,   12,     8,     2;
[5]  10,   50,  100,   100,    50,    10;
[6]   5,   30,   75,   100,    75,    30,     5;
[7]  35,  245,  735,  1225,  1225,   735,   245,   35;
[8]  14,  112,  392,   784,   980,   784,   392,  112,   14;
[9] 126, 1134, 4536, 10584, 15876, 15876, 10584, 4536, 1134, 126;

Cf. A057977, A063549, A240558 (row sums), A000007 (alternating row sums).

T := proc(n, k) n!^2 / (k! * (n - k)! * iquo(n,2)!^2 * (iquo(n,2) + 1)) end:
for n from 0 to 9 do seq(T(n, k), k = 0..n) od;

T(n, k) = binomial(n, k) * A057977(n).

cp's OEIS Frontend

A057977 GCD of consecutive central binomial coefficients: a(n) = gcd(A001405(n+1), A001405(n)).

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A274709 A statistic on orbital systems over n sectors: the number of orbitals which rise to maximum height k over the central circle.

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

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A063550 Largest number of crossing-free matchings on a set S of n points in the plane, that is, a set of floor(n/2) pairwise non-intersecting segments with endpoints in S having no endpoint in common.

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A359648 Triangle read by rows. T(n, k) = (n!)^2 / (k! * (n - k)! * (floor(n/2)!)^2 * (floor(n/2) + 1)).

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