A005566 -id:A005566 - cp's OEIS frontend

A060294 Decimal expansion of Buffon's constant 2/Pi.

6, 3, 6, 6, 1, 9, 7, 7, 2, 3, 6, 7, 5, 8, 1, 3, 4, 3, 0, 7, 5, 5, 3, 5, 0, 5, 3, 4, 9, 0, 0, 5, 7, 4, 4, 8, 1, 3, 7, 8, 3, 8, 5, 8, 2, 9, 6, 1, 8, 2, 5, 7, 9, 4, 9, 9, 0, 6, 6, 9, 3, 7, 6, 2, 3, 5, 5, 8, 7, 1, 9, 0, 5, 3, 6, 9, 0, 6, 1, 4, 0, 3, 6, 0, 4, 5, 5, 2, 1, 1, 0, 6, 5, 0, 1, 2, 3, 4, 3, 8, 2, 4, 2, 9, 1

Offset: 0

Jason Earls, Mar 28 2001

The probability P(l,d) that a needle of length l will land on a line, given a floor with equally spaced parallel lines at a distance d (>=l) apart, is (2/Pi)*(l/d). - Benoit Cloitre, Oct 14 2002
Lim_{n->infinity} z(n)/log(n) = 2/Pi, where z(n) is the expected number of real zeros of a random polynomial of degree n with real coefficients chosen from a standard Gaussian distribution (cf. Finch reference). - Benoit Cloitre, Nov 02 2003
Also the ratio of the average chord length when two points are chosen at random on a circle of radius r to the maximum possible chord length (i.e., diameter) = A088538*r / (2*r) = 2/Pi. Is there a (direct or obvious) relationship between this fact and that 2/Pi is the "magic geometric constant" for a circle (see MathWorld link)? - Rick L. Shepherd, Jun 22 2006
Blatner (1997) says that Euler found a "fascinating infinite product" for Pi involving the prime numbers, but the number he then describes does not match Pi. Switching the numerator and the denominator results in this number. - Alonso del Arte, May 16 2012
2/Pi is also the height (the ordinate y) of the geometric centroid of each arbelos (see the references and links given under A221918) with a large radius r=1 and any small ones r1 and r2 = 1 - r1, for 0 < r1 < 1. Use the integral formula given, e.g., in the MathWorld or Wikipedia centroid reference, for the two parts of the arbelos (dissected by the vertical line x = 2*r1), and then use the decomposition formula. The heights y1 and y2 of the centroids of the two parts satisfy: F1(r1)*y1(r1) = 2*r1^2*(1-r1) and F2(1-r1)*y2(1-r1) = 2*(1-r1)^2*r1. The r1 dependent area F = F1 + F2 is Pi*r1*(1-r1). (F1 and F2 are rather complicated but their explicit formulas are not needed here.) The r1 dependent horizontal coordinate x with origin at the left tip of the arbelos is x = r1 + 1/2. - Wolfdieter Lang, Feb 28 2013
Construct a quadrilateral of maximal area inside a circle. The quadrilateral is necessarily an inscribed square (with diagonals that are diameters). 2/Pi is the ratio of the square's area to the circle's area. - Rick L. Shepherd, Aug 02 2014
The expected number of real roots of a real polynomial of degree n varies as this constant times the (natural) logarithm of n, see Kac, when its coefficients are chosen from the standard uniform distribution. This may be related to Rick Shepherd's comment. - Charles R Greathouse IV, Oct 06 2014
2/Pi is also the minimum value, at x = 1/2, on (0,1) of 1/(Pi*sqrt(x*(1-x))), the nonzero piece of the probability density function for the standard arcsine distribution. - Rick L. Shepherd, Dec 05 2016
The average distance from the center of a unit-radius circle to the midpoints of chords drawn between two points that are uniformly and independently chosen at random on the circumference of the circle. - Amiram Eldar, Sep 08 2020
2/Pi <= sin(x)/x < 1 for 0 < |x| <= Pi/2 is Jordan's inequality, also known as (2/Pi) * x <= sin(x) <= x for 0 <= x <= Pi/2; this inequality was named after the French mathematician Camille Jordan (1838-1922). - Bernard Schott, Jan 07 2023
This constant 2/Pi was named after the needle experiment, described in 1777 by the French naturalist and mathematician Georges-Louis Leclerc, Comte de Buffon (1707-1788). Note that the parrot Buffon's macaw and the antelope Buffon's kob were named also after Buffon. - Bernard Schott, Jan 10 2023
2*n*log(n)/Pi is also the dominant term in the asymptotic expansion of Sum_{k=1..n-1} csc(Pi*k/n) at n tending to infinity. - Iaroslav V. Blagouchine, Apr 21 2025

			2/Pi = 0.6366197723675813430755350534900574481378385829618257949906...

David Blatner, The Joy of Pi. New York: Walker & Company (1997): 119, circle by upper right corner.
G. Buffon, Essai d'arithmétique morale. Supplément à l'Histoire Naturelle, Vol. 4, 1777.
Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, pp. 141, 539.
Steven R. Finch, Mathematical Constants II, Cambridge University Press, 2018, p. 196.
G. H. Hardy, Ramanujan: twelve lectures on subjects suggested by his life and work, AMS Chelsea Publ., Providence, RI, 2002, p. 7, eq. (1.2) and p. 105 eq. (7.4.2) with s=1/2.
Robert Kanigel, The Man Who Knew Infinity: A Life of the Genius Ramanujan, 1991.
Daniel A. Klain and Gian-Carlo Rota, Introduction to Geometric Probability, Cambridge, 1997, see Chap. 1.
Luis A. Santaló, Integral Geometry and Geometric Probability, Addison-Wesley, 1976.
David Wells, The Penguin Dictionary of Curious and Interesting Numbers. Penguin Books, NY, 1986, Revised edition 1987. See p. 53.
Robert M. Young, Excursions in Calculus, An Interplay of the Continuous and the Discrete. Dolciani Mathematical Expositions Number 13. MAA.

Harry J. Smith, Table of n, a(n) for n = 0..20000
Iaroslav V. Blagouchine and Eric Moreau, On a finite sum of cosecants appearing in various problems, Journal of Mathematical Analysis and Applications, vol. 539, no. 1, pt. 2, pp. 1-36, 2024. See p. 18.
K. S. Brown, MathPages: The Algebra of an Infinite Grid of Resistors
G. Buffon, Essai d'arithmétique morale, Supplément à l'Histoire Naturelle, Vol. 4, 1777.
Encyclopedia of Mathematics, Arcsine distribution
Boris Gourevitch, L'univers de Pi
Mark Kac, On the average number of real roots of a random algebraic equation, Bull. Amer. Math. Soc. 49:4 (1943), pp. 314-320.
MacTutor History of Mathematics archive, Georges-Louis Leclerc, Comte de Buffon.
Veikko Nevanlinna, On constants connected with the prime number theorem for arithmetic progressions, Annales Academiae Scientiarum Fennicae Ser. A. I., No. 539 (1973).
Da-Wei Niu, Jian Cao, and Feng Qi, Generalizations of Jordan's inequality and concerned relations, U.P.B. Sci. Bull., Series A, Vol. 72, Iss. 3, 2010
Herbert Solomon, Geometric Probability, SIAM, 1978, p. 152. [See average chord length comment]
Eric Weisstein's World of Mathematics, Buffon's needle problem.
Eric Weisstein's World of Mathematics, Magic Geometric Constants.
Eric Weisstein's World of Mathematics, Prime Products.
Eric Weisstein's World of Mathematics, Geometric Centroid.
Eric Weisstein's World of Mathematics, Jordan's Inequality.
Wikipedia, Buffon's needle problem.
Wikipedia, Centroid.
Wikipedia, Jordan's inequality.
Index entries for transcendental numbers.

Cf. A000796 (Pi), A088538, A154956, A082542 (numerators in an infinite product), A053300 (continued fraction without the initial 0).

Cf. A076668 (sqrt(2/Pi)).

R:= RealField(100); 2/Pi(R); // G. C. Greubel, Mar 09 2018

Digits:=100: evalf(2/Pi); # Wesley Ivan Hurt, Aug 02 2014

Mathematica
```
RealDigits[ N[ 2/Pi, 111]][[1]]
```

default(realprecision, 20080); x=20/Pi; for (n=0, 20000, d=floor(x); x=(x-d)*10; write("b060294.txt", n, " ", d)); \\ Harry J. Smith, Jul 03 2009

2/Pi = 1 - 5*(1/2)^3 + 9*((1*3)/(2*4))^3 - 13*((1*3*5)/(2*4*6))^3 ... - Jason Earls [formula corrected by Paul D. Hanna, Mar 23 2013]
The preceding formula is 2/Pi = Sum_{n>=0} (-1)^n * (4*n+1) * Product_{k=1..n} (2*k-1)^3/(2*k)^3. - Alexander R. Povolotsky, Mar 24 2013. [See the Hardy reference. - Wolfdieter Lang, Nov 13 2016]
2/Pi = Product_{n>=2} (p(n) + 2 - (p(n) mod 4))/p(n), where p(n) is the n-th prime. - Alonso del Arte, May 16 2012
2/Pi = Sum_{k>=0} ((2*k)!/(k!)^2)^3*((42*k+5)/(2^{12*k+3})) (due to Ramanujan). - L. Edson Jeffery, Mar 23 2013
Equals sinc(Pi/2). - Peter Luschny, Oct 04 2019
From A.H.M. Smeets, Apr 11 2020: (Start)
Equals Product_{i > 0} cos(Pi/2^(i+1)).
Equals Product_{i > 0} f_i(2)/2, where f_0(2) = 0, f_(i+1)(2) = sqrt(2+f_i(2)) for i >= 0; a formula by François Viète (16th century).
Note that cos(Pi/2^(i+1)) = f_i(2)/2, i >= 0. (End)
Equals lim_{n->infinity} (1/n) * Sum_{k=1..n} abs(sin(k * m)) for all nonzero integers m (conjectured). Works with cos also. - Dimitri Papadopoulos, Jul 17 2020
From Amiram Eldar, Sep 08 2020: (Start)
Equals Product_{k>=1} (1 - 1/(2*k)^2).
Equals lim_{k->oo} (2*k+1)*binomial(2*k,k)^2/2^(4*k).
Equals Sum_{k>=0} binomial(2*k,k)^2/((2*k+2)*2^(4*k)). (End)
Equals Sum_{k>=0} mu(4*k+1)/(4*k+1) (Nevanlinna, 1973). - Amiram Eldar, Dec 21 2020
Equals 1 - Sum_{n >= 1} (1/16^n) * binomial(2*n, n)^2 * 1/(2*n - 1). See Young, p. 264. - Peter Bala, Feb 17 2024
Equals binomial(0, 1/2) = binomial(0, -1/2). - Peter Luschny, Dec 05 2024
From Peter Bala, Dec 10 2024:(Start)
2/Pi =  1 - 1/(2 + 2/(1 + 6/(1 + 12/(1 + 20/(1 + ... + n*(n+1)/(1 + ...), a continued fraction representation due to Euler. See A346943.
Equals 1 - (1/2)*Sum_{n >= 0} A005566(n)*(-1/4)^n. (End)

A039648 Number of n-step self-avoiding paths on the first octant of a 3-dimensional cubic lattice, starting at the origin.

Original entry on oeis.org

1, 3, 9, 33, 123, 489, 1947, 7977, 32817, 137253, 576993, 2452071, 10468245, 45032733, 194475321, 844608567, 3680153043, 16105438515, 70677344403, 311242931097, 1373860647453, 6081635195553

Offset: 0

David W. Wilson

Cf. A039618, A005566.

a(19)-a(21) from Scott R. Shannon, Aug 14 2020

Title clarified by Sean A. Irvine, Feb 20 2021

A088855 Triangle read by rows: number of symmetric Dyck paths of semilength n with k peaks.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 2, 4, 2, 1, 1, 3, 6, 6, 3, 1, 1, 3, 9, 9, 9, 3, 1, 1, 4, 12, 18, 18, 12, 4, 1, 1, 4, 16, 24, 36, 24, 16, 4, 1, 1, 5, 20, 40, 60, 60, 40, 20, 5, 1, 1, 5, 25, 50, 100, 100, 100, 50, 25, 5, 1, 1, 6, 30, 75, 150, 200, 200, 150, 75, 30, 6, 1, 1, 6, 36, 90, 225, 300, 400, 300, 225, 90, 36, 6, 1

Offset: 1

Emeric Deutsch, Nov 24 2003

Rows 2, 4, 6, ... give A088459.
Diagonal sums are in A088518(n-1). - Philippe Deléham, Jan 04 2009
Row sums are in A001405(n). - Philippe Deléham, Jan 04 2009
Subtriangle (1 <= k <= n) of triangle T(n,k), 0 <= k <= n, read by rows, given by A101455 DELTA A056594 := [0,1,0,-1,0,1,0,-1,0,1,0,-1,0,...] DELTA [1,0,-1,0,1,0,-1,0,1,0,-1,0,1,...] where DELTA is the operator defined in A084938. - Philippe Deléham, Jan 03 2009
Also, number of symmetric noncrossing partitions of an n-set with k blocks. - Andrew Howroyd, Nov 15 2017
From Roger Ford, Oct 17 2018: (Start)
T(n,k) = t(n+2,d) where t(n,d) is the number of different semi-meander arch depth listings with n top arches and with d the depth of the deepest embedded arch.
Examples:    /\         semi-meander with 5 top arches
            //\\   /\   2 arches are at depth=0 (no covering arches)
           ///\\\ //\\  2 arches are at depth=1 (1 covering arch)
      (0)(1)(2)         1 arch is at depth=2 (2 covering arches)
       2, 2, 1 is the listing for this t(5,2)
             /\      semi-meander with 5 top arches
            /  \     (0)(1)
     /\ /\ //\/\\     3, 2  is the listing for this t(5,1)
a(6,5) = t(8,5)= 3 {2,1,1,1,2,1; 2,1,2,1,1,1; 3,1,1,1,1,1} (End)

			Triangle begins:
  1;
  1,  1;
  1,  1,  1;
  1,  2,  2,   1;
  1,  2,  4,   2,   1;
  1,  3,  6,   6,   3,    1;
  1,  3,  9,   9,   9,    3,    1;
  1,  4, 12,  18,  18,   12,    4,    1;
  1,  4, 16,  24,  36,   24,   16,    4,    1;
  1,  5, 20,  40,  60,   60,   40,   20,    5,    1;
  1,  5, 25,  50, 100,  100,  100,   50,   25,    5,    1;
  1,  6, 30,  75, 150,  200,  200,  150,   75,   30,    6,   1;
  1,  6, 36,  90, 225,  300,  400,  300,  225,   90,   36,   6,   1;
  1,  7, 42, 126, 315,  525,  700,  700,  525,  315,  126,  42,   7,  1;
  1,  7, 49, 147, 441,  735, 1225, 1225, 1225,  735,  441, 147,  49,  7, 1;
  1,  8, 56, 196, 588, 1176, 1960, 2450, 2450, 1960, 1176, 588, 196, 56, 8, 1;
  ...
a(6,2)=3 because we have UUUDDDUUUDDD, UUUUDDUUDDDD, UUUUUDUDDDDD, where
U=(1,1), D=(1,-1).

Andrew Howroyd, Table of n, a(n) for n = 1..1275
Per Alexandersson, Svante Linusson, Samu Potka, and Joakim Uhlin, Refined Catalan and Narayana cyclic sieving, arXiv:2010.11157 [math.CO], 2020.
Paul Barry, On Integer-Sequence-Based Constructions of Generalized Pascal Triangles, Journal of Integer Sequences, Vol. 9 (2006), Article 06.2.4.
Hyunsoo Cho, JiSun Huh, and Jaebum Sohn, The (s, s + d, ..., s + pd)-core partitions and the rational Motzkin paths, arXiv:2001.06651 [math.CO], 2020.
Johann Cigler, Some remarks and conjectures related to lattice paths in strips along the x-axis, arXiv:1501.04750 [math.CO], 2015-2016.
Johann Cigler, Pascal triangle, Hoggatt matrices, and analogous constructions, arXiv:2103.01652 [math.CO], 2021.
Nicolas Crampe, Julien Gaboriaud, and Luc Vinet, Racah algebras, the centralizer Z_n(sl_2) and its Hilbert-Poincaré series, arXiv:2105.01086 [math.RT], 2021.
L. Poulain d'Andecy, Centralisers and Hecke algebras in Representation Theory, with applications to Knots and Physics, arXiv:2304.00850 [math.RT], 2023. See p. 64.
Vladimir Shevelev, Several remarks on A088855, Seqfan thread, Nov 19 2017.

Cf. A005566, A018224, A056594, A084938, A088459, A101455, A209612, A247644.
Cf. A001405 (row sums), A088459, A088518 (diagonal sums).
Column 2 is A008619, column 3 is A002620, column 4 is A028724, column 5 is A028723, column 6 is A028725, column 7 is A331574.

[(&*[Binomial(Floor((n-j)/2), Floor((k-j)/2)): j in [0..1]]): k in [1..n], n in [1..15]]; // G. C. Greubel, Apr 08 2022

T[n_, k_] := Binomial[Quotient[n-1, 2], Quotient[k-1, 2]]*Binomial[ Quotient[n, 2], Quotient[k, 2]];
Table[T[n, k], {n,13}, {k,n}]//Flatten (* Jean-François Alcover, Jun 07 2018 *)

T(n,k) = binomial((n-1)\2, (k-1)\2)*binomial(n\2, k\2); \\ Andrew Howroyd, Nov 15 2017

def A088855(n,k): return product(binomial( (n-j)//2, (k-j)//2 ) for j in (0..1))
flatten([[A088855(n,k) for k in (1..n)] for n in (1..15)]) # G. C. Greubel, Apr 08 2022

T(n, k) = binomial(floor(n'), floor(k'))*binomial(ceiling(n'), ceiling(k')), where n' = (n-1)/2, k' = (k-1)/2.
G.f.: 2*u/(u*v + sqrt(x*y*u*v)) - 1, where x = 1+z+t*z, y = 1+z-t*z, u = 1-z+t*z, v = 1-z-t*z.
Triangle T(n,k), 0 <= k <= n, given by A101455 DELTA A056594 begins: 1; 0,1; 0,1,1; 0,1,1,1; 0,1,2,2,1; 0,1,2,4,2,1; 0,1,3,6,6,3,1; 0,1,3,9,9,9,3,1; ... - Philippe Deléham, Jan 03 2009
From G. C. Greubel, Apr 08 2022: (Start)
T(n, n-k+1) = T(n, k).
T(2*n-1, n) = A018224(n-1), n >= 1.
T(2*n, n) = A005566(n-1), n >= 1. (End)

Keyword:tabl added Philippe Deléham, Jan 25 2010

A060900 Number of walks of length n on square lattice, starting at origin, staying on points with x >= 0, y <= x.

Original entry on oeis.org

1, 2, 7, 21, 78, 260, 988, 3458, 13300, 47880, 185535, 680295, 2649570, 9841260, 38470380, 144263925, 565514586, 2136388436, 8392954570, 31893227366, 125515281892, 479240167224, 1888770070824, 7240285271492, 28569774314536, 109883747363600, 434040802086220

Offset: 0

David W. Wilson, May 05 2001

nonn

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; https://specfun.inria.fr/bostan/HDR.pdf

Alois P. Heinz, Table of n, a(n) for n = 0..500
A. Bostan, Computer Algebra for Lattice Path Combinatorics, Séminaire de Combinatoire Ph. Flajolet, March 28 2013.
A. Bostan, Computer Algebra for Lattice Path Combinatorics, 7th Séminaire Lotharingien de Combinatoire, Ellwangen, March 23-25, 2015.

Cf. A005566, A001700, A060897, A060898, A060899.

b:= proc(n, x, y) option remember;
      `if`(x<0 or y>x, 0, `if`(n=0, 1, add(add(
       b(n-1, x+i, y+j), j=[-1, 1]), i=[-1, 1])))
    end:
a:= n-> b(n, 0$2):
seq(a(n), n=0..30);  # Alois P. Heinz, Nov 30 2015

(* Conjectural *) a[0]=1; a[n_] := a[n] = If[EvenQ[n], (4*(3*n+1)*a[n-1])/ (3*n+2), (4*n*a[n-1])/(n+1)]; Table[a[n], {n, 0, 26}]
(* or, from 1st g.f. *) s = (HypergeometricPFQ[{-1/12, 1/4}, {2/3}, -64*x* (4*x+1)^2/(4*x-1)^4]-1)/(2*x) + O[x]^27; CoefficientList[s, x](* Jean-François Alcover, Nov 30 2015 *)

The following conjectural formula for this sequence is apparently due to Ira M. Gessel: a(0) = 1, a(2n) = a(2n-1)*(12n+2)/(3n+1), a(2n+1) = a(2n)*(4n+2)/(n+1).
G.f.: (hypergeom([ -1/12, 1/4],[2/3],-64*x*(4*x+1)^2/(4*x-1)^4)-1)/(2*x). - Mark van Hoeij, Nov 02 2009
G.f.: (T(x)-1)/(2*x) where T(x) satisfies 27*(4*x-1)^2*T^8 - 18*(4*x-1)^2*T^4 - (128*x^2+192*x+8)*T^2 - (4*x-1)^2 = 0. - Mark van Hoeij, Nov 02 2009
a(n) ~ 4^(n+1) / (sqrt(3) * Gamma(1/3) * n^(2/3)). - Vaclav Kotesovec, Sep 17 2017

Entry revised by N. J. A. Sloane at the suggestion of Doron Zeilberger, Sep 13 2007

A093768 Positive first differences of the rows of triangle A088459, which enumerates symmetric Dyck paths.

Original entry on oeis.org

1, 1, 1, 1, 2, 3, 1, 3, 8, 6, 1, 4, 15, 20, 20, 1, 5, 24, 45, 75, 50, 1, 6, 35, 84, 189, 210, 175, 1, 7, 48, 140, 392, 588, 784, 490, 1, 8, 63, 216, 720, 1344, 2352, 2352, 1764, 1, 9, 80, 315, 1215, 2700, 5760, 7560, 8820, 5292, 1, 10, 99, 440, 1925, 4950, 12375, 19800

Offset: 0

Paul D. Hanna, Apr 16 2004

Suggested by Bozydar Dubalski (slawb(AT)atr.bydgoszcz.pl). Related to walks on a square lattice: main diagonal forms A005558, secondary diagonals form A005559, A005560, A005561, A005562, A005563.

Apparently row-reversed version of A052174. - R. J. Mathar, Feb 03 2025

			1;
1, 1;
1, 2, 3;
1, 3, 8, 6;
1, 4, 15, 20, 20;
1, 5, 24, 45, 75, 50;
1, 6, 35, 84, 189, 210, 175;

G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened

Cf. A088459, A005558-A005562, A005563 (column 3), A005564 (column 4), A005565 (column 5), A005566 (row sums).

A093768 := proc(n,k)
    if k = 0 then
        A088459(n,k);
    else
        A088459(n,k)-A088459(n,k-1);
    end if;
end proc:
seq(seq(A093768(n,k),k=0..n-1),n=1..10) ; # R. J. Mathar, Apr 02 2017

T[n_, k_] := Binomial[n + 1, Ceiling[k/2]]*Binomial[n, Floor[k/2]] - Binomial[n + 1, Ceiling[(k - 1)/2]]*Binomial[n, Floor[(k - 1)/2]]; Table[T[n, k], {n, 0, 10}, {k, 0, n}] // Flatten (* G. C. Greubel, Oct 25 2017 *)

{T(n,k) =binomial(n+1,ceil(k/2))*binomial(n,floor(k/2)) -binomial(n+1,ceil((k-1)/2))*binomial(n,floor((k-1)/2))}

T(n, k) = C(n+1, ceiling(k/2))*C(n, floor(k/2)) - C(n+1, ceiling((k-1)/2))*C(n, floor((k-1)/2)) for n>=k>=0.

A060897 Number of walks of length n on square lattice, starting at origin, staying in first and third quadrants.

Original entry on oeis.org

1, 4, 12, 44, 144, 528, 1808, 6676, 23536, 87568, 315136, 1180680, 4314560, 16263896, 60138816, 227899484, 850600944, 3238194560, 12177384544, 46542879384, 176110444736, 675431779856, 2568878867200, 9882068082112, 37747540858240, 145593279888736, 558190182662144

Offset: 0

David W. Wilson, May 05 2001

nonn

Is there a formula analogous to the (conjectured) formula for A060900?

Could be broken into the number of walks that are constrained to a quadrant and the number that cross the origin. (I.e., 2*A005566(n) + 2*A005566(n-2)*A005568(1) + 2*A005566(n-4)*A005568(2) + ... + All terms that cross the origin twice + three times + ... + Cross floor(n/2) times.) - Benjamin Phillabaum, Mar 13 2011

Alois P. Heinz, Table of n, a(n) for n = 0..750 (first 251 terms from Sean A. Irvine)

Cf. A005566, A005568, A001700, A060898, A060899, A060900.

\\ here B is A005566 and C is aerated A005568 as g.f.'s.
B(n)={sum(n=0, n, x^n*binomial(n, n\2)*binomial(n+1, (n+1)\2), O(x*x^n))}
C(n)={sum(n=0, (n+1)\2, x^(2*n)*binomial(2*n,n)*binomial(2*n+2,n+1)/((n+1)*(n+2)), O(x*x^n))}
seq(n) = {Vec( 1 + 2*(B(n)-1)/(2-C(n)) )} \\ Andrew Howroyd, Jan 05 2023

G.f.: 1 + 2*(B(x)-1)/(2 - C(x^2)) where B(x) is the g.f. of A005566 and C(x) is the g.f. of A005568. - Andrew Howroyd, Jan 05 2023

A060898 Number of walks of length n on square lattice, starting at origin, staying in first, second and third quadrants.

Original entry on oeis.org

1, 4, 14, 54, 200, 776, 2940, 11466, 43980, 172170, 665544, 2612764, 10154144, 39949000, 155864280, 614260062, 2403739140, 9486263092, 37209147800, 147012850512, 577741491404, 2284848892872, 8993216244896, 35595538140656, 140288753584200, 555662416386840

Offset: 0

David W. Wilson, May 05 2001

Is there a formula analogous to the (conjectured) formula for A060900?

Alois P. Heinz, Table of n, a(n) for n = 0..1000
M. Bousquet-Mélou, Plane lattice walks avoiding a quadrant, arXiv:1511.02111 [math.CO], 2015.
Mireille Bousquet-Mélou, Square lattice walks avoiding a quadrant, Journal of Combinatorial Theory, Series A, Elsevier, 2016, Special issue for the 50th anniversary of the journal, 144, pp. 37-79. Also . See App. A.
Kilian Raschel, Amélie Trotignon, On walks avoiding a quadrant, arXiv:1807.08610 [math.CO], 2018.

Cf. A005566, A001700, A060897, A060899, A060900.

A241530 a(n) = binomial(n,floor(n/2))binomial(n+1,floor(n/2+1/2))(1+floor(n/2))/(1+2*floor(n/2)).

Original entry on oeis.org

1, 2, 4, 12, 36, 120, 400, 1400, 4900, 17640, 63504, 232848, 853776, 3171168, 11778624, 44169840, 165636900, 625739400, 2363904400, 8982836720, 34134779536, 130332794592, 497634306624, 1907598175392, 7312459672336, 28124844893600, 108172480360000

Offset: 0

Peter Luschny, Apr 25 2014

Alois P. Heinz, Table of n, a(n) for n = 0..1000

Cf. A000894, A002894, A005566.

Row n=3 of A275784.

A241530 := n -> binomial(n,iquo(n,2))*binomial(n+1,iquo(n+1,2))
*(1+iquo(n,2))/(1+2*iquo(n,2)); seq(A241530(n), n=0..26);
# second Maple program:
a:= proc(n) option remember; `if`(n<2, 2^n,
     ((8*n-4)*a(n-1)+16*(n-1)*(n-2)*a(n-2))/(n*(n+1)))
    end:
seq(a(n), n=0..30);  # Alois P. Heinz, Aug 10 2016

CoefficientList[Series[(-EllipticE[16 x^2] + (1 + 4 x) EllipticK[16 x^2])/(2Pi x), {x, 0, 20}], x] (* Benedict W. J. Irwin, Aug 15 2016 *)
Table[Binomial[n, #] Binomial[n + 1, Floor[(n + 1)/2]] (1 + #)/(1 + 2 #) &@ Floor[n/2], {n, 0, 26}] (* Michael De Vlieger, Aug 15 2016 *)

a(n) = ((8*n-4)*a(n-1)+16*(n-1)*(n-2)*a(n-2))/(n*(n+1)) for n>=2, a(n) = 2^n for n<2. - Alois P. Heinz, Apr 25 2014
G.f.: ((1+4*x)*K(4*x) - E(4*x))/(2*Pi*x), where K and E are the complete elliptic integrals of the first and second kind, respectively, with modulus k = 4*x. - Benedict W. J. Irwin, Aug 15 2016
From Wolfdieter Lang, Sep 06 2016 (Start):
The preceding g.f. can be rewritten as ((1+4*x)*F(1/2,1/2;1;(4*x)^2) -
   F(-1/2,1/2;1;(4*x)^2))/(4*x), where F is the hypergyometric function F(a,b;c;z).
This leads to the bisection a(2*k) = ((2*k)!)^2/k!^4 = A002894(k) and a(2*k+1) = 2*(2*k)!*(2*k+1)!/((k+1)*k!^4) = 2*A000894(k), for k >= 0.
(End)

A060899 Number of walks of length n on square lattice, starting at origin, staying on points with x+y >= 0.

Original entry on oeis.org

1, 2, 8, 24, 96, 320, 1280, 4480, 17920, 64512, 258048, 946176, 3784704, 14057472, 56229888, 210862080, 843448320, 3186360320, 12745441280, 48432676864, 193730707456, 739699064832, 2958796259328, 11342052327424

Offset: 0

David W. Wilson, May 05 2001

The number of lattice paths consisting of 2*n steps either (1,1) or (1,-1) that return to the x-axis only at times that are a multiple of 4. - Peter Bala, Jan 02 2020

Harry J. Smith, Table of n, a(n) for n=0..200
Paul Barry, The Central Coefficients of a Family of Pascal-like Triangles and Colored Lattice Paths, J. Int. Seq., Vol. 22 (2019), Article 19.1.3.
Alin Bostan, Computer Algebra for Lattice Path Combinatorics, Séminaire de Combinatoire Ph. Flajolet, March 28 2013.
Alin Bostan, Andrew Elvey Price, Anthony John Guttmann, and Jean-Marie Maillard, Stieltjes moment sequences for pattern-avoiding permutations, arXiv:2001.00393 [math.CO], 2020.
Math Overflow, Combinatorial proof for the number of lattice paths that return to the axis only at times that are a multiple of 4, 2012.

Cf. A005566, A001700, A060897, A060898, A060900.

Cf. A001405.

Table[2^n Binomial[n,Floor[n/2]],{n,0,30}] (* Harvey P. Dale, Oct 15 2017 *)

{ for (n=0, 200, write("b060899.txt", n, " ", 2^n*binomial(n, n\2)); ) } \\ Harry J. Smith, Jul 14 2009

a(n) = 2^n*binomial(n, [n/2]);
G.f.: (sqrt((1+4*x)/(1-4*x))-1)/4/x. - Vladeta Jovovic, Apr 28 2003
E.g.f.: BesselI(0, 4*x)+BesselI(1, 4*x). - Vladeta Jovovic, Apr 28 2003
a(n) = 4^n*sum{k=0..n, C(n,k)C(k)/(-2)^k}, with C(n)=A000108(n). - Paul Barry, Dec 28 2006
(n+1)*a(n) -4*a(n-1) +16*(-n+1)*a(n-2)=0. - R. J. Mathar, Nov 24 2012
a(n) = (-4)^n*hypergeom([3/2,-n],[2],2). - Peter Luschny, Apr 26 2016
Sum_{n>=0} a(n)/6^n = 3/phi = A134973. - Peter McNair, Apr 30 2022
In general, for k>4, Sum_{n>=0} a(n)/k^n = (sqrt((k+4)/(k-4)) - 1) * k/4. - Vaclav Kotesovec, May 13 2022
From Amiram Eldar, May 14 2022: (Start)
Sum_{n>=0} 1/a(n) = 16*asin(1/4)/(3*sqrt(15)) + 4/3.
Sum_{n>=0} (-1)^n/a(n) = 4/5 - 16*asin(1/4)/(5*sqrt(15)). (End)

A064044 Square array read by antidiagonals of number of length k walks on an n-dimensional hypercubic lattice starting at the origin and staying in the nonnegative part.

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 0, 2, 2, 1, 0, 3, 6, 3, 1, 0, 6, 18, 12, 4, 1, 0, 10, 60, 51, 20, 5, 1, 0, 20, 200, 234, 108, 30, 6, 1, 0, 35, 700, 1110, 624, 195, 42, 7, 1, 0, 70, 2450, 5460, 3760, 1350, 318, 56, 8, 1, 0, 126, 8820, 27405, 23480, 9770, 2556, 483, 72, 9, 1, 0, 252

Offset: 0

Henry Bottomley, Aug 23 2001

E.g.f. of row n equals ( besseli(0,2*y) + y*besseli(1,2*y) )^n. - Paul D. Hanna, Apr 07 2005

			Rows start:
1, 0,  0,   0,    0,     0,      0, ...
1, 1,  2,   3,    6,    10,     20, ...
1, 2,  6,  18,   60,   200,    700, ...
1, 3, 12,  51,  234,  1110,   5460, ...
1, 4, 20, 108,  624,  3760,  23480, ...
1, 5, 30, 195, 1350,  9770,  73300, ...
1, 6, 42, 318, 2556, 21480, 187140, ...

Alois P. Heinz, Antidiagonals n = 0..140, flattened
R. K. Guy, Catwalks, sandsteps and Pascal pyramids, J. Integer Sequences, Vol. 3 (2000), Article #00.1.6

Rows include A000007, A001405, A005566, A064036. Columns include A000012, A001477, A002378, A064043. Cf. A064045.

a:= proc(n, k) option remember; `if`(n=0, `if`(k=0, 1, 0),
       add(binomial(k, j)*binomial(j, floor(j/2))
       *a(n-1, k-j), j=0..k))
    end:
seq(seq(a(n,d-n), n=0..d), d=0..12);  # Alois P. Heinz, May 06 2014

a[n_, k_] := a[n, k] = If[n == 0, If[k == 0, 1, 0], Sum[Binomial[k, j]*Binomial[j, Floor[j/2]]*a[n-1, k-j], {j, 0, k}]]; Table[Table[a[n, d-n], {n, 0, d}], {d, 0, 12}] // Flatten (* Jean-François Alcover, Feb 26 2015, after Alois P. Heinz *)

{T(n,k)=local(X=x+x*O(x^n),Y=y+y*O(y^k)); k!*polcoeff(polcoeff(1/(1-X*besseli(0,2*Y)-X*Y*besseli(1,2*Y)),n,x),k,y)} /* Hanna */

a(n,k) = Sum{j=0..k} C(k,j) B(j) a(n-1,k-j) where B(j) = C(j,[j/2]) = A001405(j) with a(0,0) = 1 and a(0,k) = 0 for k>0.

E.g.f: 1/(1 - x*besseli(0, 2*y) - x*y*besseli(1, 2*y)). - Paul D. Hanna, Apr 07 2005

cp's OEIS Frontend

A060294 Decimal expansion of Buffon's constant 2/Pi.

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A039648 Number of n-step self-avoiding paths on the first octant of a 3-dimensional cubic lattice, starting at the origin.

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A088855 Triangle read by rows: number of symmetric Dyck paths of semilength n with k peaks.

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A060900 Number of walks of length n on square lattice, starting at origin, staying on points with x >= 0, y <= x.

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A093768 Positive first differences of the rows of triangle A088459, which enumerates symmetric Dyck paths.

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A060897 Number of walks of length n on square lattice, starting at origin, staying in first and third quadrants.

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A060898 Number of walks of length n on square lattice, starting at origin, staying in first, second and third quadrants.

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A241530 a(n) = binomial(n,floor(n/2))*binomial(n+1,floor(n/2+1/2))*(1+floor(n/2))/(1+2*floor(n/2)).

A241530 a(n) = binomial(n,floor(n/2))binomial(n+1,floor(n/2+1/2))(1+floor(n/2))/(1+2*floor(n/2)).