A337900 -id:A337900 - cp's OEIS frontend

A060150 a(0) = 1; for n > 0, binomial(2n-1, n-1)^2.

1, 1, 9, 100, 1225, 15876, 213444, 2944656, 41409225, 590976100, 8533694884, 124408576656, 1828114918084, 27043120090000, 402335398890000, 6015361252737600, 90324408810638025, 1361429497505672100, 20589520178326522500, 312321918272897610000

Offset: 0

N. J. A. Sloane, Apr 10 2001

Number of square lattice walks that start at (0,0) and end at (1,0) after 2n-1 steps, free to pass through (1,0) at intermediate steps. - Steven Finch, Dec 20 2001
Number of paths of length n connecting two neighboring nodes in optimal chordal graph of degree 4, G(2*d(G)^2+2*d(G)+1,2d(G)+1), of diameter d(G). - B. Dubalski (dubalski(AT)atr.bydgoszcz.pl), Feb 05 2002
a(n) is the number of ways to place n red balls and n blue balls into n distinguishable boxes with no restrictions on the number of balls put in a box. - Geoffrey Critzer, Jul 08 2013
The number of square lattice walks of n steps that start at the origin and end at (k,0) is zero if n-k is odd and [binomial(n,(n-k)/2)]^2 if n-k is even. - R. J. Mathar, Sep 28 2020

R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics, 1994 Addison-Wesley company, Inc.
A. P. Prudnikov, Yu. A. Brychkov and O.I. Marichev, "Integrals and Series", Volume 1: "Elementary Functions", New York, Gordon and Breach Science Publishers, 1986-1992, Eq. (5.1.29.2)
K. A. Ross and C. R. B. Wright, Discrete Mathematics, 1992 Prentice Hall Inc.

Harry J. Smith, Table of n, a(n) for n = 0..200
R. Bacher, Meander algebras

Cf. A002894, A135389, A337900, A337901, A337902.

seq(coeff(series(EllipticK(4*sqrt(x))/(2*Pi) + 3/4, x=0, n+1), x, n), n=0..30);  # Mark van Hoeij, Apr 30 2013

Table[Binomial[2n-1,n]^2,{n,0,19}] (* Geoffrey Critzer, Jul 08 2013 *)

PARI
```
a(n)=if(n<2, 1, binomial(2*n-1,n-1)^2)
```

for (n=0, 200, if (n==0, a=1, a=binomial(2*n - 1, n - 1)^2); write("b060150.txt", n, " ", a)) \\ Harry J. Smith, Jul 02 2009

a(n) = A088218(n)^2.
a(n) = A002894(n)/4 for n>0.
G.f.: 1 + (1/AGM(1, sqrt(1-16*x))-1)/4. - Michael Somos, Dec 12 2002
G.f. = 1 + (K(16x)-1)/4 = 1 + Sum_{k>0} q^k/(1+q^(2k)) where K(16x) is the complete Elliptic integral of the first kind at 16x=k^2 and q is the nome. - Michael Somos, May 09 2005
G.f.: 1 + x*3F2((1, 3/2, 3/2); (2, 2))(16*x). - Olivier Gérard, Feb 16 2011
E.g.f.: Sum_{n>0} a(n)*x^(2n-1)/(2n-1)! = BesselI(0, 2x)*BesselI(1, 2x) . - Michael Somos, Jun 22 2005
D-finite with recurrence n^2*a(n) -4*(2*n-1)^2*a(n-1)=0. - R. J. Mathar, Jul 26 2014
From Seiichi Manyama, Oct 19 2016: (Start)
Let the number of multisets of length k on n symbols be denoted by ((n, k)) = binomial(n+k-1, k).
a(n) = (Sum_{0 <= k <= n} binomial(n, k)^2 * ((2*n, n - k)))/3 for n > 0. (End)
a(n) ~ 4^(2*n-1)/(Pi*n). - Ilya Gutkovskiy, Oct 19 2016
For n >= 1, a(n) = 1/n * Sum_{k = 0..n-1} (n + 2*k)*binomial(n+k-1, k)^2 = ( 1/(4*n) * Sum_{k = 0..n} (n + 2*k)*binomial(-n+k-1, k)^2 )^2. - Peter Bala, Nov 02 2024

A135389 Number of walks of length 2*n+2 from origin to (1,1) in a square lattice.

Original entry on oeis.org

2, 24, 300, 3920, 52920, 731808, 10306296, 147232800, 2127513960, 31031617760, 456164781072, 6749962774464, 100445874620000, 1502052155856000, 22557604697766000, 340044833169460800, 5143178101688094600

Offset: 0

Stefan Hollos (stefan(AT)exstrom.com), Dec 11 2007

a(n) is the number of walks of length 2n+2 in an infinite square lattice that begin at the origin and end at (1,1) using steps (1,0), (-1,0), (0,1), (0,-1).

			G.f. = 2 + 24*x + 300*x^2 + 3920*x^3 + 731808*x^4 + 10306296*x^5 + ... - _Michael Somos_, Oct 17 2019

S. Hollos and R. Hollos, Lattice Paths and Walks.

Cf. A002894, A060150, A337900-A337902.

series( 2*hypergeom([3/2, 3/2],[3],16*x), x=0, 20);  # Mark van Hoeij, Apr 06 2013

Table[Binomial[2n + 2, n] Binomial[2n + 2, n + 1], {n, 0, 19}] (* Alonso del Arte, Apr 06 2013 *)

a(n) = binomial(2n+2,n) * binomial(2n+2,n+1) = A001791(n+1)*A000984(n+1).
G.f.: 2*2F1(3/2,3/2; 3; 16*x). - Mark van Hoeij, Apr 06 2013
D-finite with recurrence n*(n+2)*a(n) -4*(2*n+1)^2*a(n-1)=0. - R. J. Mathar, Jul 14 2013
E.g.f.: Sum_{n>0} a(n-1) * x^(2*n)/(2*n)! = BesselI(1, 2*x)^2. - Michael Somos, Oct 17 2019

A337902 The number of walks of length 2n+1 on the square lattice that start from the origin (0,0) and end at the vertex (2,1).

Original entry on oeis.org

3, 50, 735, 10584, 152460, 2208492, 32207175, 472780880, 6982113996, 103673813880, 1546866469148, 23179817220000, 348690679038000, 5263441096145400, 79698007774092375, 1210159553338375200, 18422202264818467500, 281089726445607849000

Offset: 1

R. J. Mathar, Sep 29 2020

			a(1)=3 represents 3 walks of length 3: RRU, URR and RUR.

Cf. A002894 (at (0,0)), A060150 (at (1,0)), A135389 (at (1,1)), A337900 (at (2,0)), A337901 (at (3,0))

a(n) =  binomial(2*n+1,n-1)*binomial(2*n+1,n) = A002054(n)*A001700(n).
G.f.: 3*x*3F2(2,5/2,5/2; 3,4; 16*x).
D-finite with recurrence  (n-1)*(n+2)*(n+1)*a(n) -4*n*(2*n+1)^2*a(n-1)=0.
A135389(n) = 2*A060150(n+1) +2*a(n).

A337901 The number of walks of length 2n+1 on the square lattice that start from the origin (0,0) and end at the vertex (3,0).

Original entry on oeis.org

1, 25, 441, 7056, 108900, 1656369, 25050025, 378224704, 5712638724, 86394844900, 1308887012356, 19868414760000, 302198588499600, 4605510959127225, 70321771565375625, 1075697380745222400, 16483023079048102500, 252980753801047064100, 3888662839165553120100

Offset: 1

R. J. Mathar, Sep 29 2020

Cf. A002894 (end at (0,0)), A060150 (end at (1,0)), A135389 (end at (1,1)), A337900 (at (2,0)), A337902 (at(2,1))

a(n) = [A002054(n)]^2.
G.f.: x*4F3(2,2,5/2,5/2; 1,4,4; 16*x).
D-finite with recurrence (n+2)^2*(n-1)^2*a(n) -4*n^2*(2*n+1)^2*a(n-1)=0.

A378060 a(n) = binomial(n, floor((n-1)/2))^2.

Original entry on oeis.org

0, 1, 1, 9, 16, 100, 225, 1225, 3136, 15876, 44100, 213444, 627264, 2944656, 9018009, 41409225, 130873600, 590976100, 1914762564, 8533694884, 28210561600, 124408576656, 418151049316, 1828114918084, 6230734868736, 27043120090000, 93271169290000, 402335398890000

Offset: 0

Peter Luschny, Dec 03 2024

Number of walks of length n with unit steps in all four directions (NSWE), starting at the origin and ending on the y-axis, never going below the x-axis and the end point having a positive height.

			The 16 walks of length 4: NNNN, NNNS, NNSN, NNEW, NNWE, NSNN, NENW, NEWN, NWNE, NWEN, ENNW, ENWN, EWNN, WNNE, WNEN, WENN.

Cf. A060150 (odd bisection), A337900 (even bisection), A037952, A378061.

# Generates the walks (for illustration only).
function aCount(n::Int)
    a = [""]
    c = 0
    for w in a
        if length(w) == n
            if (count('N', w) != count('S', w) && count('W', w) == count('E', w))
                c += 1
                # println(w)
            end
        else
            for j in "NSEW"
                u = string(w, j)
                if count('N', u) >= count('S', u)
                   push!(a, u)
    end end end end
    return c
end
println([aCount(n) for n in 0:11])

a := n -> binomial(n, iquo(n+1, 2) - 1)^2: seq(a(n), n = 0..27);
a := proc(n) option remember; if n < 2 then n else ((32*n^5 - 48*n^4 + 16*n^2)*a(n - 2) + (8*n^4 - 20*n^2)*a(n - 1))/(2*(n - 1)^2*(n - 1/2)*(n + 2)^2) fi end:
# Alternative:
egf := BesselI(0, 2*x)^2 + BesselI(1, 2*x)*BesselI(0, 2*x)*(1 - 1/x):
ser := series(egf, x, 29): seq(n!*coeff(ser, x, n), n = 0..27);

Array[Binomial[#, Floor[(# + 1)/2] - 1]^2 &, 28, 0] (* Michael De Vlieger, Dec 04 2024 *)

a(n) = n!*[x^n] (BesselI(0, 2*x)^2 + BesselI(1, 2*x)*BesselI(0, 2*x)*(1 - 1/x)).
a(n) = [x^n] (((8*x^2 + 2*x)*EllipticK(4*x) - Pi*(1 + x) + 2*EllipticE(4*x))/(4*x^2*Pi)).
a(n) = [x^n] (x*hypergeom([1,3/2,3/2], [2,2], 16*x^2) + x^2*hypergeom([3/2,3/2,2,2], [1,3,3], 16*x^2)).
a(n) = Sum_{k=0..n} (-1)^(n-k+N)*C(n-k, N)*C(n, k)*C(n+k, k), where N = floor((n-1)/2) and C = binomial.
Recurrence: a(n) = ((32*n^5 - 48*n^4 + 16*n^2)*a(n - 2) + (8*n^4 - 20*n^2)*a(n - 1))/(2*(n - 1)^2*(n - 1/2)*(n + 2)^2).
a(n) = Sum_{k=1..n} A378061(n, k).

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A060150 a(0) = 1; for n > 0, binomial(2n-1, n-1)^2.

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A135389 Number of walks of length 2*n+2 from origin to (1,1) in a square lattice.

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A337902 The number of walks of length 2n+1 on the square lattice that start from the origin (0,0) and end at the vertex (2,1).

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A337901 The number of walks of length 2n+1 on the square lattice that start from the origin (0,0) and end at the vertex (3,0).

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A378060 a(n) = binomial(n, floor((n-1)/2))^2.

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