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 1-7 of 7 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

A145600 a(n) is the number of walks from (0,0) to (0,1) that remain in the upper half-plane y >= 0 using (2*n - 1) unit steps either up (U), down (D), left (L) or right (R).

Original entry on oeis.org

1, 8, 75, 784, 8820, 104544, 1288287, 16359200, 212751396, 2821056160, 38013731756, 519227905728, 7174705330000, 100136810390400, 1409850293610375, 20002637245262400, 285732116760449700
Offset: 1

Views

Author

Peter Bala, Oct 14 2008

Keywords

Comments

Cf. A000891, which enumerates walks in the upper half-plane starting and finishing at the origin. See also A145601, A145602 and A145603. This sequence is the central column taken from triangle A145596, which enumerates walks in the upper half-plane starting at the origin and finishing on the horizontal line y = 1.

Examples

			a(2) = 8: the 8 walks from (0,0) to (0,1) of three steps are
UDU, UUD, URL, ULR, RLU, LRU, RUL and LUR.

References

  • M. Dukes and Y. Le Borgne, Parallelogram polyominoes, the sandpile model on a complete bipartite graph, and a q,t-Narayana polynomial, Journal of Combinatorial Theory, Series A, Volume 120, Issue 4, May 2013, Pages 816-842. - From N. J. A. Sloane, Feb 21 2013

Links

Crossrefs

Cf. A000891, A145596, A145601, A145602, A145603.

Programs

  • Maple
    a(n) := 1/n*binomial(2*n,n+1)*binomial(2*n,n-1);
seq(a(n),n = 1..19);

Formula

a(n) = 1/n*binomial(2*n,n+1)*binomial(2*n,n-1).
a(n) = A135389(n-1)/(n+1). - R. J. Mathar, Jul 14 2013
D-finite with recurrence (n+1)^2*a(n) -4*n*(5*n-1)*a(n-1) +16*(2*n-3)^2*a(n-2)=0. - R. J. Mathar, Jul 14 2013

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

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

Views

Author

R. J. Mathar, Sep 29 2020

Keywords

Examples

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

Crossrefs

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

Formula

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

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.

A378070 a(n) = binomial(n - 1, ceiling(n/2)) * binomial(n - 1, ceiling(n/2) - 1).

Original entry on oeis.org

1, 0, 1, 2, 9, 24, 100, 300, 1225, 3920, 15876, 52920, 213444, 731808, 2944656, 10306296, 41409225, 147232800, 590976100, 2127513960, 8533694884, 31031617760, 124408576656, 456164781072, 1828114918084, 6749962774464, 27043120090000, 100445874620000, 402335398890000
Offset: 0

Views

Author

Peter Luschny, Dec 13 2024

Keywords

Links

Crossrefs

Cf. A007318, A060150 (even bisection), A135389 (odd bisection), A378060.

Programs

  • Maple
    a :=  n -> binomial(n-1, floor((n+1)/2))*binomial(n-1, floor((n+1)/2)-1);
seq(a(n), n = 0..27);
  • Mathematica
    A378070[n_] := Binomial[n - 1, #]*Binomial[n - 1, # - 1] & [Ceiling[n/2]];
Array[A378070, 30, 0] (* Paolo Xausa, Dec 14 2024 *)

Formula

a(n) = binomial(n - 1, floor(n/2) - 1) * binomial(n - 1, ceiling(n/2) - 1).

A227169 a(n) = 3*((2*n+2)!)^2 / (n!*(n+1)!*(n+2)!*(n+3)!).

Original entry on oeis.org

1, 6, 45, 392, 3780, 39204, 429429, 4907760, 58023108, 705264040, 8772399636, 111263122656, 1434941066000, 18775651948200, 248797110637125, 3333772874210400, 45115597383228900, 615974564891763000, 8477309210264363700, 117511846058893572000
Offset: 0

Views

Author

Karol A. Penson, Jul 12 2013

Keywords

Links

Programs

  • Maple
    seq(3*((2*n+2)!)^2/(n!*(n+1)!*(n+2)!*(n+3)!),n=0..15);
  • Mathematica
    Table[3*((2*n + 2)!)^2/(n!*(n + 1)!*(n + 2)!*(n + 3)!), {n, 0, 20}] (* T. D. Noe, Jul 12 2013 *)
  • Sage
    def a(n): return 3*(n+1)*(n+2)^2*(n+3)^3*gamma(2*n+3)^2/gamma(n+4)^4
[a(n) for n in (0..16)]  # Peter Luschny, Jul 12 2013

Formula

In Maple notation,
ogf(z) = 3/(4*z^2) +(1/12288)*(-98304*z^2-2048*z+512)*EllipticK(4*sqrt(z))/(z^3*Pi) +(1/12288)*(-20480*z-512)*EllipticE(4*sqrt(z))/(z^3*Pi);
egf(z)=hypergeom([3/2, 3/2, 2], [1, 3, 4], 16*z), a 3F3 hypergeometric function.
Integral representation as the n-th moment of a signed function w(x) of bounded variation, 0<=x<=16: a(n) = Integral_{x=0..16}x^n*w(x), n>=0, where w(x) is the Meijer G function, w(x) = -3*MeijerG([[0], [2, 3]], [[1/2, 1/2], [1]], (1/16)*x)/Pi, satisfying w(16)=w(0)=0, w(x)<0 for x < 0.47.
The above Meijer G function cannot be represented by any other special function.
(n+3)*(n+2)*a(n) -18*(n+1)^2*a(n-1) +8*(2*n-1)^2*a(n-2)=0. - R. J. Mathar, Jul 14 2013
a(n) = 3*A135389(n)/((n+2)*(n+3)) = 3*A145600(n+1)/(n+3). - R. J. Mathar, Jul 14 2013
Showing 1-7 of 7 results.

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

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