A056040 -id:A056040 - cp's OEIS frontend

A100071 a(n) = n * binomial(n-1, floor((n-1)/2)) = n * max_{i=0..n} binomial(n-1, i).

0, 1, 2, 6, 12, 30, 60, 140, 280, 630, 1260, 2772, 5544, 12012, 24024, 51480, 102960, 218790, 437580, 923780, 1847560, 3879876, 7759752, 16224936, 32449872, 67603900, 135207800, 280816200, 561632400, 1163381400, 2326762800

Offset: 0

Paul Barry, Nov 02 2004

Old name: An inverse Chebyshev transform of n.
Hankel transform is (-1)^n*n*2^(n-1), A085750. This is the inverse binomial transform of -n. - Paul Barry, Jan 11 2007
Corollary 3 of the Farhi reference mentions this sequence. - Roger L. Bagula, Nov 08 2009
Number of UDUD's in all length n+3 left factors of Dyck paths (here U=(1,1) and D=(1,-1)). Example: a(2)=2 because in (UDUD)U, UDUUD, UDUUU, UUDDU, U(UDUD), UUDUU, UUUDD, UUUDU, UUUUD, and UUUUU we have a total of two UDUDs (shown between parentheses).  Also number of UUDD's in all length n+3 left factors of Dyck paths (here U=(1,1) and D=(1,-1)). Example: a(2)=2 because in UDUDU, UDUUD, UDUUU, (UUDD)U, UUDUD, UUDUU, U(UUDD), UUUDU, UUUUD, and UUUUU we have a total of two UUDDs (shown between parentheses). - Emeric Deutsch, Jun 19 2011
Apparently the number of long ascents in all symmetric Dyck (n+1)-paths. - David Scambler, Aug 17 2012
Beginning with the least positive term multiple of an odd prime p (which is a(p)), we have exactly p+1 consecutive terms multiple of p. - Vladimir Shevelev, Aug 17 2012
Apparently also the count of 'unmatched symbols' in the binary strings of length n (see A008314). - Wouter Meeussen, May 26 2013

T. D. Noe, Table of n, a(n) for n = 0..1000
Ruggero Bandiera and Florian Schaetz, Eulerian idempotent, pre-Lie logarithm and combinatorics of trees, arXiv:1702.08907 [math.CO], 2017. See p. 34.
F. Disanto, A. Frosini, and S. Rinaldi, Square involutions, J. Int. Seq., Vol. 14 (2011), Article 11.3.5.
Bakir Farhi, An identity involving the least common multiple of binomial coefficients and its application, The American Mathematical Monthly, Vol. 116, No. 9 (2009), pp. 836-839, see p. 838; arXiv preprint, arXiv:0906.2295 [math.NT], 2009.
Nikita Gogin and Mika Hirvensalo, On the Moments of Squared Binomial Coefficients, (2020).
D. Nečas, Elementary models of low-pressure plasma polymerisation into nanofibrous mats, Phys. Scr. (2025) Vol. 100, 055601. See p. 10.
Helmut Prodinger, Dispersed Dyck paths revisited, arXiv:2402.13026 [math.CO], 2024.

Cf. A134757, A008314.

[n*Binomial(n-1, Floor((n-1)/2)): n in [0..35]]; // Vincenzo Librandi, Sep 14 2015

swing := n -> n!/iquo(n,2)!^2:
A100071 := n -> swing(n)*(n/2)^(n-1 mod 2):
seq(A100071(i),i=0..30); # Peter Luschny, Aug 31 2011

Table[(Floor[n/2] + Ceiling[n/2] + 1)!/(Floor[n/2]!*Ceiling[n/2]!), {n, 1, 40}] (* Stefan Steinerberger, Nov 04 2008 *)
Table[If[n == 0, 0, n*Binomial[n - 1, Floor[(n - 1)/2]]], {n, 0, 30}] (* Roger L. Bagula, Nov 08 2009 *);
Table[ Tr[ Table[Count[match[-1 + 2*IntegerDigits[n, 2, k]], 0], {n, 2^(k - 1), 2^k - 1}]], {k, 16}] (* function 'match' see A008314; Wouter Meeussen, May 26 2013 *)

a(n) = n * binomial(n-1, (n-1)\2); \\ Michel Marcus, Sep 14 2015

def A100071(n):
    f = factorial(n)/factorial(n//2)^2
    return f if is_odd(n) else f*(n/2)
[A100071(n) for n in (0..50)]  # Peter Luschny, Aug 17 2012

G.f.: 2*x*(1 - sqrt(1 - 4*x^2))/(sqrt(1 - 4*x^2)*(sqrt(1 - 4*x^2) + 2*x - 1)^2).
G.f.: (1/sqrt(1 - 4*x^2))*x*c(x^2)/(1 - x*c(x^2))^2.
a(n) = Sum_{k = 0..floor(n/2)} binomial(n,k)*(n - 2*k).
Sum_{k = 0..floor(n/2)} binomial(n-k,k)*(-1)^k*a(n-2k) = 1.
From Paul Barry, Jan 11 2007: (Start)
a(n) = n*binomial(n-1, floor((n-1)/2));
a(n) = Sum_{k = 0..n} binomial(n,k)*2^(n-k)*binomial(2*k-2, k-1)*(-1)^(k-1). (End)
Starting (1, 2, 6, 12, ...), = inverse binomial transform of A134757: (1, 3, 11, 37, 123, 401, ...). - Gary W. Adamson, Nov 08 2007
a(n) = a(n-1)*n/floor(n/2) for n > 0. - Reinhard Zumkeller, Jan 20 2008
G.f.: x/((1 - 2*x)*sqrt(1 - 4*x^2)). - Paul Barry, Apr 25 2008
a(n) = (floor(n/2) + ceiling(n/2) + 1)!/(floor(n/2)! * ceiling(n/2)!). - Stefan Steinerberger, Nov 04 2008
a(n) = A056040(n)*(n/2)^((n-1) mod 2). - Peter Luschny, Aug 31 2011
Asymptotic: a(n) ~ b(n) where b(n) = ceiling(2^(n-1)*sqrt(2*n-(-1)^n)/sqrt(Pi)). b(n) is also a lower bound of a(n) and an upper bound of 2^(n-1). With corollary 3 from Bakir Farhi (see reference) lcm(1,2,...,n) >= a(n) >= b(n) >= 2^(n-1). - Peter Luschny, Aug 17 2012
a(n) = n for n < 3, a(n) = 4*a(n-2) + 2*a(n-1)/(n-1) for n >= 3. - Alexander R. Povolotsky, Aug 17 2012
E.g.f.: x*(BesselI(0,2*x) + BesselI(1,2*x)). - Peter Luschny, Aug 19 2012
a(n) = (-1)^(n*(n+1)/2) * Sum_{k = 0..n} (-1)^k*k*binomial(n,k)^2. - Peter Bala, Jul 25 2016
a(n) = n!/(floor((n-1)/2)!*ceiling((n-1)/2)!). See the Bandiera link. - Michel Marcus, Feb 28 2017
D-finite with recurrence (-n+1)*a(n) + 2*a(n-1) + 4*(n-1)*a(n-2) = 0. - R. J. Mathar, Aug 09 2017
From Amiram Eldar, Mar 10 2022: (Start)
Sum_{n>=1} 1/a(n) = Pi/sqrt(3).
Sum_{n>=1} (-1)^(n+1)/a(n) = Pi/(3*sqrt(3)). (End)

Name changed, using part of a comment from Paul Barry, by Peter Luschny, Aug 17 2012

A046212 First numerator and then denominator of central elements of Leibniz's Harmonic Triangle.

Original entry on oeis.org

1, 1, 1, 6, 1, 30, 1, 140, 1, 630, 1, 2772, 1, 12012, 1, 51480, 1, 218790, 1, 923780, 1, 3879876, 1, 16224936, 1, 67603900, 1, 280816200, 1, 1163381400, 1, 4808643120, 1, 19835652870, 1, 81676217700, 1, 335780006100, 1, 1378465288200, 1

Offset: 1

Mohammad K. Azarian

L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 83, Problem 25.

Cf. A003506.

Cf. A002457.

a(2n+1) = A056040(2n+1) = A100071(2n+1). - M. F. Hasler, Jan 25 2012

More terms from James Sellers, Dec 13 1999

A055773 a(n) = Product_{p in P_n} where P_n = {p prime, n/2 < p <= n }.

Original entry on oeis.org

1, 1, 2, 6, 3, 15, 5, 35, 35, 35, 7, 77, 77, 1001, 143, 143, 143, 2431, 2431, 46189, 46189, 46189, 4199, 96577, 96577, 96577, 7429, 7429, 7429, 215441, 215441, 6678671, 6678671, 6678671, 392863, 392863, 392863, 14535931, 765049, 765049, 765049

Offset: 0

Labos Elemer, Jul 12 2000

nonn

Old name: Product of primes p for which p divides n! but p^2 does not (i.e. ord_p(n!)=1). - Dion Gijswijt (gijswijt(AT)science.uva.nl), Jan 07 2007
Squarefree part of n! divided by gcd(Q,F), where Q is the largest square divisor and F is the squarefree part of n!. - Labos Elemer, Jul 12 2000
a(1) = 1, a(n) = n*a(n-1) if n is a prime else a(n) = least integer multiple of a(n-1)/n. - Amarnath Murthy, Apr 29 2004
Let P(i) denote the primorial number A034386(i). Then a(n) = P(n)/P(floor(n/2)). - Peter Luschny, Mar 05 2011
Letting H(n) = 1 + 1/2 + ... + 1/n denote the n-th harmonic number, it is known that a(n) is equal to the denominator (in lowest terms) of H(n)^2*n! for n >= 6 (see below example). - John M. Campbell, Mar 27 2016
For all n satisfying 6 <= n < 897, a(n) = A130087(n). - John M. Campbell, Mar 27 2016
It is also known that a(n) is equal to lcm^2(1, 2, ..., n)/gcd(lcm^2(1, 2, ..., n), n!). - John M. Campbell, Apr 04 2016

			n = 13, P_n = {7, 11, 13}, a(13) = 7*11*13 = 1001.
Letting n = 14, the denominator (in lowest terms) of H(n)^2*n! = 131803989435744/143 is a(14)=143. - _John M. Campbell_, Mar 27 2016

Vincenzo Librandi, Table of n, a(n) for n = 0..200 (corrected by Michel Marcus, Jan 19 2019)
J. M. Campbell et al., A problem involving the product prod_{k=1..n} k^mu(k), where mu denotes the Möbius function, Mathematics Stack Exchange (2016).
F. J. van de Bult, D. C. Gijswijt, J. P. Linderman, N. J. A. Sloane and Allan Wilks, A Slow-Growing Sequence Defined by an Unusual Recurrence, J. Integer Sequences, Vol. 10 (2007), #07.1.2.
Index entries for sequences related to Gijswijt's sequence

Cf. A000188, A008833, A007913, A055229, A055231 (for n), A055071, A055204, A055230, A094299, A094302, A193477, A130087.

a := n -> mul(k,k=select(isprime,[$iquo(n,2)+1..n])); # Peter Luschny, Jun 20 2009
A055773 := n -> numer(n!/iquo(n,2)!^4); # Peter Luschny, Jul 30 2011

Table[Numerator[n!/Floor[n/2]!^4], {n, 0, 40}] (* Michael De Vlieger, Mar 27 2016 *)

q=1;for(n=2,41,print1(q,",");q=if(isprime(n),q*n,q/gcd(q,n))) \\ Klaus Brockhaus, May 02 2004

a(n) = k=1;forprime(p=nextprime(n\2+1),precprime(n),k=k*p);k \\ Klaus Brockhaus, May 02 2004

a(n) = prod(i=primepi(n/2)+1,primepi(n),prime(i)) \\ John M. Campbell, Mar 27 2016

from math import prod
from sympy import primerange
def A055773(n): return prod(primerange((n>>1)+1,n+1)) # Chai Wah Wu, Apr 13 2024

a(n) = numerator(A056040(n)^2/n!).
a(n) = numerator(A056040(n)/floor(n/2)!^2).
a(n) = numerator(n!/floor(n/2)!^4). - Peter Luschny, Jul 30 2011
a(n) = product of primes p such that n/2 < p <= n. - Klaus Brockhaus, May 02 2004
a(n) = A055204(n)/A055230(n) = A055231(n!) = A007913(n!)/A055229(n!).
a(n) = Product_{i=pi(n/2)+1..pi(n)} prime(i), where pi denotes the prime counting function and prime(i) denotes the i-th prime number. - John M. Campbell, Mar 27 2016

Entry revised by N. J. A. Sloane, Jan 07 2007

Simpler definition based on a comment of Klaus Brockhaus, set offset to 0 and prepended 1 to data. - Peter Luschny, Mar 09 2013

A232500 Oscillating orbitals over n sectors (nonpositive values indicating there exist none).

Original entry on oeis.org

-1, -1, 0, 0, 2, 10, 10, 70, 42, 378, 168, 1848, 660, 8580, 2574, 38610, 10010, 170170, 38896, 739024, 151164, 3174444, 587860, 13520780, 2288132, 57203300, 8914800, 240699600, 34767720, 1008263880, 135727830, 4207562730, 530365050, 17502046650, 2074316640

Offset: 0

Peter Luschny, Jan 05 2014

A planar orbital system is a family of concentric circles in a plane divided into n sectors. An orbital is a closed path consisting of arcs on these circles such that at each boundary of a sector the path jumps to the next inner or outer circle. One exception is allowed: if n is odd the path might continue on the same circle, but just once. After fixing one circle as the central circle there are three types of orbitals: a high orbital is always above the central circle, a low orbital is always below the central circle, and an oscillating orbital which is neither a high nor a low orbital. The number of all orbitals is A056040(n), the number of high orbitals, which is the same as the number of low orbitals, is A057977(n), and the number of oscillating orbitals is this a(n) (for n >= 4).

Peter Luschny, Illustrating swinging orbitals.

Cf. A056040, A057977.

f := (z/(1-4*z^2)-3-1/z+1/z^2)/sqrt(1-4*z^2)-1/z^2+1/z;
seq(coeff(series(f, z, n+2), z, n), n=0..19);
g := (1+x)*BesselI(0, 2*x)-2*(1+1/x)*BesselI(1, 2*x);
seq(n!*coeff(series(g,x,n+2),x,n), n=0..19);

sf[n_] := n!/Quotient[n, 2]!^2; a[n_] := sf[n]*(1-2/(Quotient[n, 2]+1)); Table[a[n], {n, 0, 40}] (* Jean-François Alcover, Feb 11 2015 *)
nxt[{n_,a_}]:={n+1,If[EvenQ[n],a(n+1),a (4(n-1))/((n-3)(n+3))]}; Join[{-1,-1,0,0,2},NestList[nxt,{5,10},40][[;;,2]]] (* Harvey P. Dale, Dec 16 2024 *)

a(n) = n!/(n\2)!^2*(n\2-1)/(n\2+1) \\ Charles R Greathouse IV, Jul 30 2016

def A232500():
    r, n = 1, 0
    while True:
        yield r*(n//2-1)/(n//2+1)
        n += 1
        r *= 4/n if is_even(n) else n
a = A232500(); [next(a) for i in range(36)]

O.g.f.: (z/(1-4*z^2) - 3 - 1/z + 1/z^2)/sqrt(1-4*z^2) - 1/z^2 + 1/z.
E.g.f.: (1+x)*BesselI(0, 2*x)-2*(1+1/x)*BesselI(1, 2*x).
a(n) = (n!/k!^2)*(k-1)/(k+1) where k = floor(n/2).
Recurrence: If n > 4 then a(n) = a(n-1)*n if n is odd else a(n-1)*4*(n-2)/((n-4)*(n+2)).
a(n) = A056040(n) * (1 - 2/(floor(n/2) + 1)).
a(n) = A056040(n) - 2*A057977(n).
Asymptotic: log(a(n)) ~ (n*log(4) - log(Pi) - (-1)^n*(log(n/2) + 1/(2*n)))/2 + log(1 - 8/(2*n + 3 + (-1)^n)) for n >= 4.
D-finite with recurrence: +(n+2)*a(n) -n*a(n-1) +(-11*n+2)*a(n-2) +(9*n-16)*a(n-3) +20*(2*n-5)*a(n-4) +20*(-n+3)*a(n-5) +48*(-n+5)*a(n-6)=0. - R. J. Mathar, Feb 21 2020

A060632 a(n) = 2^wt(floor(n/2)) (i.e., 2^A000120(floor(n/2)), or A001316(floor(n/2))).

Original entry on oeis.org

1, 1, 2, 2, 2, 2, 4, 4, 2, 2, 4, 4, 4, 4, 8, 8, 2, 2, 4, 4, 4, 4, 8, 8, 4, 4, 8, 8, 8, 8, 16, 16, 2, 2, 4, 4, 4, 4, 8, 8, 4, 4, 8, 8, 8, 8, 16, 16, 4, 4, 8, 8, 8, 8, 16, 16, 8, 8, 16, 16, 16, 16, 32, 32, 2, 2, 4, 4, 4, 4, 8, 8, 4, 4, 8, 8, 8, 8, 16, 16, 4, 4, 8, 8, 8, 8, 16, 16, 8, 8, 16, 16, 16, 16, 32

Offset: 0

Avi Peretz (njk(AT)netvision.net.il), Apr 15 2001

nonn

Number of conjugacy classes in the symmetric group S_n that have odd number of elements.
Also sequence A001316 doubled.
Number of even numbers whose binary expansion is a child of the binary expansion of n. - Nadia Heninger and N. J. A. Sloane, Jun 06 2008
First differences of A151566. Sequence gives number of toothpicks added at the n-th generation of the leftist toothpick sequence A151566. - N. J. A. Sloane, Oct 20 2010
The Fi1 and Fi1 triangle sums, see A180662 for their definitions, of Sierpiński's triangle A047999 equal this sequence. - Johannes W. Meijer, Jun 05 2011
Also number of odd entries in n-th row of triangle of Stirling numbers of the first kind. - Istvan Mezo, Jul 21 2017

			a(3) = 2 because in S_3 there are two conjugacy classes with odd number of elements, the trivial conjugacy class and the conjugacy class of transpositions consisting of 3 elements: (12),(13),(23).
From _Omar E. Pol_, Oct 12 2011 (Start):
Written as a triangle:
1,
1,
2,2,
2,2,4,4,
2,2,4,4,4,4,8,8,
2,2,4,4,4,4,8,8,4,4,8,8,8,8,16,16,
2,2,4,4,4,4,8,8,4,4,8,8,8,8,16,16,4,4,8,8,8,8,16,16,8,...
(End)

I. G. MacDonald: Symmetric functions and Hall polynomials Oxford: Clarendon Press, 1979. Page 21.

Indranil Ghosh, Table of n, a(n) for n = 0..65536 (terms 0..1000 from Harry J. Smith)
David Applegate, Omar E. Pol and N. J. A. Sloane, The Toothpick Sequence and Other Sequences from Cellular Automata, Congressus Numerantium, Vol. 206 (2010), 157-191. [There is a typo in Theorem 6: (13) should read u(n) = 4.3^(wt(n-1)-1) for n >= 2.]
Christina Talar Bekaroğlu, Analyzing Dynamics of Larger than Life: Impacts of Rule Parameters on the Evolution of a Bug's Geometry, Master's thesis, Calif. State Univ. Northridge (2023). See p. 92.
N. J. A. Sloane, Catalog of Toothpick and Cellular Automata Sequences in the OEIS
Index entries for sequences related to toothpick sequences

Cf. A000120, A001316, A139251, A151566, A160407.

a000120:=func< n | &+Intseq(n, 2) >; [ 2^a000120(Floor(n/2)): n in [0..100] ]; // Klaus Brockhaus, Oct 15 2010

A060632 := proc(n) local k; add(binomial(n,2*k) mod 2, k=0..floor(n/2)); end: seq(A060632(n),n=0..94); # edited by Johannes W. Meijer, May 28 2011
A060632 := n -> 2^add(i, i = convert(iquo(n,2), base, 2)); # Peter Luschny, Jun 30 2011
A060632 := n -> igcd(2^n, n! / iquo(n,2)!^2);  # Peter Luschny, Jun 30 2011

a[n_] := 2^DigitCount[Floor[n/2], 2, 1]; Table[a[n], {n, 0, 94}] (* Jean-François Alcover, Feb 25 2014 *)

for (n=0, 1000, write("b060632.txt", n, " ", sum(k=0, floor(n/2), binomial(n, 2*k) % 2)) ) \\ Harry J. Smith, Sep 14 2009

a(n)=2^hammingweight(n\2) \\ Charles R Greathouse IV, Feb 06 2017

def A060632(n):
    return 2**bin(n/2)[2:].count("1") # Indranil Ghosh, Feb 06 2017

a(n) = sum{k=0..floor(n/2), C(n, 2k) mod 2} - Paul Barry, Jan 03 2005, Edited by Harry J. Smith, Sep 15 2009
a(n) = gcd(A056040(n), 2^n). - Peter Luschny, Jun 30 2011
G.f.: (1 + x) * Product_{k>=0} (1 + 2*x^(2^(k+1))). - Ilya Gutkovskiy, Jul 19 2019

More terms from James Sellers, Apr 16 2001
Edited by N. J. A. Sloane, Jun 06 2008; Oct 11 2010
a(0) = 1 added by N. J. A. Sloane, Sep 14 2009
Formula corrected by Harry J. Smith, Sep 15 2009

A081125 a(n) = n! / floor(n/2)!.

Original entry on oeis.org

1, 1, 2, 6, 12, 60, 120, 840, 1680, 15120, 30240, 332640, 665280, 8648640, 17297280, 259459200, 518918400, 8821612800, 17643225600, 335221286400, 670442572800, 14079294028800, 28158588057600, 647647525324800, 1295295050649600

Offset: 0

Paul Barry, Mar 07 2003

Product of the largest parts in the partitions of n+1 into exactly two parts, n > 0. - Wesley Ivan Hurt, Jan 26 2013 (Clarified on Apr 20 2016)

			a(3) = 6 since 3+1 = 4 has two partitions into two parts, (3,1) and (2,2), and the product of the largest parts is 6. - _Wesley Ivan Hurt_, Jan 26 2013 (Clarified on Apr 20 2016)

Vincenzo Librandi, Table of n, a(n) for n = 0..400
Peter Luschny, Die schwingende Fakultät und Orbitalsysteme, August 2011.
Index to divisibility sequences.

Cf. A004526, A056040, A081123, A000407 (bisection), A001813 (bisection).

[Factorial(n)/(Factorial(Floor(n/2))): n in [0..30]]; // Vincenzo Librandi, Sep 13 2011

Method 1)  a:=n->n!/floor(n/2)!; seq(a(k),k=0..40); # Wesley Ivan Hurt, Jun 03 2013
Method 2)  with(combinat, numbperm); seq(numbperm(k, floor((k+1)/2)), k = 0..40); # Wesley Ivan Hurt, Jun 06 2013

Table[n!/Floor[n/2]!, {n, 0, 30}] (* Wesley Ivan Hurt, Apr 20 2016 *)

a(n)=n!/(n\2)! \\ Charles R Greathouse IV, Sep 13 2011

from sympy import rf
def A081125(n): return rf((m:=n+1>>1)+(n+1&1),m) # Chai Wah Wu, Jul 22 2022

def a(n): return rising_factorial(ceil(n/2),floor(n/2))
[a(n) for n in range(26)]  # Peter Luschny, Oct 09 2013

E.g.f.: (1+x)*exp(x^2). - Vladeta Jovovic, Sep 24 2003
From Peter Luschny, Aug 07 2009: (Start)
a(n) = sqrt(n!*n$) where n$ denotes the swinging factorial (A056040).
a(n) = 2^n Gamma((n+1+(n mod 2))/2)/sqrt(Pi). (End)
E.g.f.: E(0) where E(k) = 1 + x/(1 - x/(x + (k+1)/E(k+1))) ; (continued fraction, 3rd kind, 3-step). - Sergei N. Gladkovskii, Sep 20 2012
G.f.: G(0) where G(k) =  1 + x*(2*k+1)/(1 - 2*x/(2*x + 1/G(k+1))); (continued fraction, 3-step). - Sergei N. Gladkovskii, Nov 18 2012
D-finite with recurrence a(n) +2*a(n-1) -2*n*a(n-2) +4*(-n+2)*a(n-3) = 0. - R. J. Mathar, Nov 26 2012
From Wesley Ivan Hurt, Jun 06 2013: (Start)
a(n) = n!/(n-floor((n+1)/2))!.
a(n) = Product_{i = ceiling(n/2)..(n-1)} i. [Note: empty product = 1]
a(n) = P( n, floor((n+1)/2) ), where P(n,k) are the number of k-permutations of n objects. (End)
a(n) = n$*floor(n/2)! where n$ denotes the swinging factorial (A056040). - Peter Luschny, Oct 28 2013
From Amiram Eldar, Mar 10 2022: (Start)
Sum_{n>=0} 1/a(n) = 1 + (3/2)*exp(1/4)*sqrt(Pi)*erf(1/2).
Sum_{n>=0} (-1)^n/a(n) = 1 - (1/2)*exp(1/4)*sqrt(Pi)*erf(1/2). (End)

A241477 Triangle read by rows, number of orbitals classified with respect to the first zero crossing, n>=1, 1<=k<=n.

Original entry on oeis.org

1, 0, 2, 2, 2, 2, 0, 4, 0, 2, 6, 12, 4, 2, 6, 0, 12, 0, 4, 0, 4, 20, 60, 12, 12, 12, 4, 20, 0, 40, 0, 12, 0, 8, 0, 10, 70, 280, 40, 60, 36, 24, 40, 10, 70, 0, 140, 0, 40, 0, 24, 0, 20, 0, 28, 252, 1260, 140, 280, 120, 120, 120, 60, 140, 28, 252, 0, 504, 0

Offset: 1

Peter Luschny, Apr 23 2014

For the combinatorial definitions see A232500. An orbital w over n sectors has its first zero crossing at k if k is the smallest j such that the partial sum(1<=i<=j, w(i))) = 0, where w(i) are the jumps of the orbital represented by -1, 0, 1.

			[1], [ 1]
[2], [ 0,  2]
[3], [ 2,  2,  2]
[4], [ 0,  4,  0,  2]
[5], [ 6, 12,  4,  2,  6]
[6], [ 0, 12,  0,  4,  0, 4]
[7], [20, 60, 12, 12, 12, 4, 20]

Row sums: A056040.

Cf. A232500.

A241477 := proc(n, k)
  if n = 0 then 1
elif k = 0 then 0
elif irem(n, 2) = 0 and irem(k, 2) = 1 then 0
elif k = 1 then (n-1)!/iquo(n-1,2)!^2
else 2*(n-k)!*(k-2)!/iquo(k,2)/(iquo(k-2,2)!*iquo(n-k,2)!)^2
  fi end:
for n from 1 to 9 do seq(A241477(n, k), k=1..n) od;

T[n_, k_] := Which[n == 0, 1, k == 0, 0, Mod[n, 2] == 0 && Mod[k, 2] == 1,  0, k == 1, (n-1)!/Quotient[n-1, 2]!^2, True, 2*(n-k)!*(k-2)!/Quotient[k, 2]/(Quotient[k-2, 2]!*Quotient[n-k, 2]!)^2];
Table[T[n, k], {n, 1, 12}, {k, 1, n}] // Flatten (* Jean-François Alcover, Jun 20 2018, from Maple *)

def A241477_row(n):
    if n == 0: return [1]
    Z = [0]*n; T = [0] if is_odd(n) else []
    for i in (1..n//2): T.append(-1); T.append(1)
    for p in Permutations(T):
        i = 0; s = p[0]
        while s != 0: i += 1; s += p[i];
        Z[i] += 1
    return Z
for n in (1..9): A241477_row(n)

If n is even and k is odd then T(n, k) = 0 else if k = 1 then T(n, 1) = A056040(n-1) else T(n, k) = 2*A057977(k-2)*A056040(n-k).
T(n, n) = A241543(n).
T(n+1, 1) = A126869(n).
T(2*n, 2*n) = |A002420(n)|.
T(2*n+1, 1) = A000984(n).
T(2*n+1, n+1) = A241530(n).
T(2*n+2, 2) = A028329(n).
T(4*n, 2*n) = |A010370(n)|.
T(4*n, 4*n) = |A024491(n)|.
T(4*n+1, 1) = A001448(n).
T(4*n+1, 2*n+1) = A002894(n).

A018224 a(n) = binomial(n, floor(n/2))^2 = A001405(n)^2.

Original entry on oeis.org

1, 1, 4, 9, 36, 100, 400, 1225, 4900, 15876, 63504, 213444, 853776, 2944656, 11778624, 41409225, 165636900, 590976100, 2363904400, 8533694884, 34134779536, 124408576656, 497634306624, 1828114918084, 7312459672336, 27043120090000, 108172480360000, 402335398890000

Offset: 0

N. J. A. Sloane

a(n) is also the number of rooted two-vertex (or, dually, two-face) regular planar maps of valency n+1. - Valery A. Liskovets, Oct 19 2005
If A is a random matrix in USp(4) (4 X 4 complex matrices that are unitary and symplectic), then a(n)=(-1)^n*E[(tr(A^4))^n]. - Andrew V. Sutherland, Apr 01 2008
Number of square lattice walks with unit steps in all four directions (NSWE), starting at the origin, ending on the y-axis, and never going below the x-axis. Row sums of A378061. - Peter Luschny, Dec 08 2024

			The 9 lattice walks defined in the comments: 'NNN', 'NNS', 'NSN', 'NWE', 'NEW', 'WNE', 'WEN', 'ENW', 'EWN'.

Stephanie Anderson, Table for n, a(n) for n = 0..200
Alin Bostan, Computer Algebra for Lattice Path Combinatorics, Slides, Séminaire de Combinatoire Ph. Flajolet, March 28 2013.
Alin Bostan, Computer Algebra for Lattice Path Combinatorics, Synthesis, Séminaire de Combinatoire Ph. Flajolet, June 06 2013.
Alin Bostan, Calcul Formel pour la Combinatoire des Marches [The text is in English], Habilitation à Diriger des Recherches, Laboratoire d'Informatique de Paris Nord, Université Paris 13, December 2017.
Alin Bostan, Frédéric Chyzak, Mark van Hoeij, Manuel Kauers, and Lucien Pech, Hypergeometric expressions for generating functions of walks with small steps in the quarter plane, Eur. J. Comb. 61, 242-275 (2017).
M. Bousquet, G. Labelle and P. Leroux, Enumeration of planar two-face maps, Discrete Math., vol. 222 (2000), 1-25.
Kiran S. Kedlaya and Andrew V. Sutherland, Hyperelliptic curves, L-polynomials and random matrices, arXiv:0803.4462 [math.NT], 2008-2010.
Helmut Prodinger, Two New Identities Involving the Catalan Numbers: A classical approach, arXiv:1911.07604 [math.CO], 2019.

Bisections are A002894 and A060150.

Cf. A000108, A001405, A056040, A113182, A138545, A378061.

s := x -> (1+x)*EllipticK(x)/(x*Pi/2)-1/x:
seq(4^n*coeff(series(s(x),x,n+2),x,n),n=0..23); # Peter Luschny, Oct 14 2015

(* Note that Mathematica uses a different definition of the EllipticK function. *)
CoefficientList[Series[(-Pi + (2 + 8 x) EllipticK[16 x^2])/(4 Pi x), {x,0,23}], x] (* Peter Luschny, Oct 14 2015 *)
Table[Binomial[n,Floor[n/2]]^2,{n,0,30}] (* Harvey P. Dale, Dec 02 2022 *)

vector(50, n, n--; binomial(n, n\2)^2) \\ Altug Alkan, Oct 14 2015

E.g.f.: BesselI(0, 2*x)*(BesselI(0, 2*x)+BesselI(1, 2*x)). - Vladeta Jovovic, Jun 12 2005
G.f. (1+1/(4*x))*hypergeom([1/2, 1/2],[1],16*x^2)-1/(4*x). - Mark van Hoeij, Oct 13 2009
a(n) = (n!/(floor(n/2)!*floor((n+1)/2)!))^2. - Peter Luschny, Apr 29 2014
a(n) = A056040(n) * A056040(n+1) / (n+1). - Peter Luschny, Apr 29 2014
a(n) = 4^n*[x^n]((1+x)*EllipticK(x)/(x*Pi/2)-1/x). - Peter Luschny, Oct 14 2015
a(n) ~ 4^n*((2*n+3)/(2*n+1))^((-1)^n/2)/((n+1)*Pi/2). - Peter Luschny, Oct 14 2015
a(n) = Sum_{k=0..n} (-1)^k*binomial(n,k)*C(k)*binomial(2*n-2*k,n-k) where C(k) are Catalan numbers (A000108), see Prodinger. - Michel Marcus, Nov 19 2019
From Peter Bala, Jul 03 2023: (Start)
Right hand side of the binomial sum identity (1/2)*Sum_{k = 0..n+1} (-1)^k*4^(n+1-k)*binomial(n+1,k)*binomial(n+k,k)*binomial(2*k,k) = a(n).
a(n) = (1/2)*4^(n+1) * hypergeom([n+1, -n-1, 1/2], [1, 1], 1).
P-recursive:
(2*n - 1)*(n + 1)^2*a(n) = 4*(2*n^2 - 1)*a(n-1) + 16*(2*n + 1)*(n - 1)^2*a(n-2) with a(0) = a(1) = 1. (End)

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))

A092287 a(n) = Product_{j=1..n} Product_{k=1..n} gcd(j,k).

Original entry on oeis.org

1, 1, 2, 6, 96, 480, 414720, 2903040, 5945425920, 4334215495680, 277389791723520000, 3051287708958720000, 437332621360674939863040000, 5685324077688774218219520000, 15974941971638268369709427589120000, 982608696336737613503095822614528000000000

Offset: 0

N. J. A. Sloane, based on a suggestion from Leroy Quet, Feb 03 2004

nonn

Conjecture: Let p be a prime and let ordp(n,p) denote the exponent of the highest power of p that divides n. For example, ordp(48,2)=4, since 48=3*(2^4). Then we conjecture that the prime factorization of a(n) is given by the formula: ordp(a(n),p) = (floor(n/p))^2 + (floor(n/p^2))^2 + (floor(n/p^3))^2 + .... Compare this to the de Polignac-Legendre formula for the prime factorization of n!: ordp(n!,p) = floor(n/p) + floor(n/p^2) + floor(n/p^3) + .... This suggests that a(n) can be considered as generalization of n!. See A129453 for the analog for a(n) of Pascal's triangle. See A129454 for the sequence defined as a triple product of gcd(i,j,k). - Peter Bala, Apr 16 2007
The conjecture is correct. - Charles R Greathouse IV, Apr 02 2013
a(n)/a(n-1) = n, n >= 1, if and only if n is noncomposite, otherwise a(n)/a(n-1) = n * f^2, f > 1. - Daniel Forgues, Apr 07 2013
Conjecture: For a product over a rectangle, f(n,m) = Product_{j=1..n} Product_{k=1..m} gcd(j,k), a factorization similar to the one given above for the square case takes place: ordp(f(n,m),p) = floor(n/p)*floor(m/p) + floor(n/p^2)*floor(m/p^2) + .... By way of directly computing the values of f(n,m), it can be verified that the conjecture holds at least for all 1 <= m <= n <= 200. - Andrey Kaydalov, Mar 11 2019

T. D. Noe, Table of n, a(n) for n = 0..67

Cf. A003989, A018806, A051190, A090494, A129365, A129439, A129453, A129454, A129455, A224479, A224497.

[n eq 0 select 1 else (&*[(&*[GCD(j,k): k in [1..n]]): j in [1..n]]): n in [0..30]]; // G. C. Greubel, Feb 07 2024

f := n->mul(mul(igcd(j,k),k=1..n),j=1..n);

a[0] = 1; a[n_] := a[n] = n*Product[GCD[k, n], {k, 1, n-1}]^2*a[n-1]; Table[a[n], {n, 0, 15}] (* Jean-François Alcover, Apr 16 2013, after Daniel Forgues *)

h(n,p)=if(nCharles R Greathouse IV, Apr 02 2013

def A092287(n):
    R = 1
    for p in primes(n+1) :
        s = 0; r = n
        while r > 0 :
            r = r//p
            s += r*r
        R *= p^s
    return R
[A092287(i) for i in (0..15)]  # Peter Luschny, Apr 10 2013

Also a(n) = Product_{k=1..n} Product_{j=1..n} lcm(1..floor(min(n/k, n/j))).
From Daniel Forgues, Apr 08 2013: (Start)
Recurrence: a(0) := 1; for n > 0: a(n) := n * (Product_{j=1..n-1} gcd(n,j))^2 * a(n-1) = n * A051190(n)^2 * a(n-1).
Formula for n >= 0: a(n) = n! * (Product_{j=1..n} Product_{k=1..j-1} gcd(j,k))^2. (End)
a(n) = n! * A224479(n)^2 (the last formula above).
a(n) = n$ * A224497(n)^4, n$ the swinging factorial A056040(n). - Peter Luschny, Apr 10 2013

Recurrence formula corrected by Daniel Forgues, Apr 07 2013

cp's OEIS Frontend

A100071 a(n) = n * binomial(n-1, floor((n-1)/2)) = n * max_{i=0..n} binomial(n-1, i).

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A046212 First numerator and then denominator of central elements of Leibniz's Harmonic Triangle.

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A055773 a(n) = Product_{p in P_n} where P_n = {p prime, n/2 < p <= n }.

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A232500 Oscillating orbitals over n sectors (nonpositive values indicating there exist none).

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A060632 a(n) = 2^wt(floor(n/2)) (i.e., 2^A000120(floor(n/2)), or A001316(floor(n/2))).

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A081125 a(n) = n! / floor(n/2)!.

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