cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-4 of 4 results.

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

Original entry on oeis.org

1, 1, 9, 100, 1225, 15876, 213444, 2944656, 41409225, 590976100, 8533694884, 124408576656, 1828114918084, 27043120090000, 402335398890000, 6015361252737600, 90324408810638025, 1361429497505672100, 20589520178326522500, 312321918272897610000
Offset: 0

Views

Author

N. J. A. Sloane, Apr 10 2001

Keywords

Comments

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

References

  • 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.

Links

Crossrefs

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

Programs

  • Maple
    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
  • Mathematica
    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)
  • PARI
    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

Formula

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

Views

Author

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

Keywords

Comments

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).

Examples

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

Links

Crossrefs

Cf. A002894, A060150, A337900-A337902.

Programs

  • Maple
    series( 2*hypergeom([3/2, 3/2],[3],16*x), x=0, 20);  # Mark van Hoeij, Apr 06 2013
  • Mathematica
    Table[Binomial[2n + 2, n] Binomial[2n + 2, n + 1], {n, 0, 19}] (* Alonso del Arte, Apr 06 2013 *)

Formula

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

A337900 The number of walks of length 2n on the square lattice that start from the origin (0,0) and end at the vertex (2,0).

Original entry on oeis.org

1, 16, 225, 3136, 44100, 627264, 9018009, 130873600, 1914762564, 28210561600, 418151049316, 6230734868736, 93271169290000, 1401915345465600, 21147754404155625, 320042195924198400, 4857445984927644900, 73916947787011560000, 1127482124965160372100
Offset: 1

Views

Author

R. J. Mathar, Sep 29 2020

Keywords

Examples

			a(2) = 16 counts the walks RRRL, RRLR, RLRR, LRRR, RRUD, RRDU, RDRU, RURD, RUDR, RDUR, URRD, DRRU, URDR, DRUR, UDRR, DURR of length 4.

Links

Crossrefs

Cf. A002894 (at (0,0)), A060150 (at (1,0)), A135389 (at (1,1)), A337901 (at (3,0)), A337902 (at (2,1)).
Cf. A001791.

Programs

  • Maple
    egf := BesselI(0, 2*x)*BesselI(2, 2*x): ser := series(egf, x, 40):
seq((2*n)!*coeff(ser, x, 2*n), n = 1..19);  # Peter Luschny, Dec 05 2024

Formula

a(n) = [A001791(n)]^2.
G.f.: x*4F3(3/2, 3/2, 2, 2; 1, 3, 3; 16*x).
D-finite with recurrence (n-1)^2*(n+1)^2*a(n) - 4*n^2*(2*n-1)^2*a(n-1) = 0.
a(n) = (2n)!*[x^(2n)] BesselI(0, 2x)*BesselI(2, 2x). - Peter Luschny, Dec 05 2024

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

Views

Author

R. J. Mathar, Sep 29 2020

Keywords

Crossrefs

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

Formula

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.
Showing 1-4 of 4 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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