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.

User: Theodore M. Mishura

Theodore M. Mishura's wiki page.

Theodore M. Mishura has authored 2 sequences.

A265132 Number of nonintersecting (or self-avoiding) rook paths joining opposite corners of an n X n grid that move in 3 or fewer cardinal directions.

Original entry on oeis.org

1, 2, 12, 108, 1180, 15300, 234374, 4190872, 86080572, 1999951380, 51874664446, 1486016035944, 46596167540806, 1587429536107688, 58385852010664650, 2305843009058576432, 97322383750732656572, 4371823119475059457716, 208254700595813407930382
Offset: 1

Views

Author

Theodore M. Mishura, Dec 02 2015

Keywords

Comments

This sequence counts all joining paths that move in one of the following ways: UP and RIGHT only; UP, RIGHT, and LEFT only; UP, RIGHT, and DOWN only.

Crossrefs

Cf. A007764.

Programs

  • Magma
    [2*(n+1)^n-Binomial(2*n,n): n in [0..20]]; // Vincenzo Librandi, Dec 03 2015
  • Mathematica
    Table[2 (n + 1)^n - Binomial[2 n, n], {n, 0, 18}] (* Michael De Vlieger, Dec 02 2015 *)
  • PARI
    a(n) = 2*(n+1)^n - binomial(2*n,n); \\ Altug Alkan, Dec 03 2015

Formula

a(n) = 2*(n+1)^n - C(2*n,n).

A257888 Number of nonintersecting (or self-avoiding) rook paths of length 2n+2 joining opposite corners of an n X n grid.

Original entry on oeis.org

4, 36, 224, 1200, 5940, 28028, 128128, 572832, 2519400, 10943240, 47070144, 200880160, 851809140, 3592795500, 15085939200, 63102895680, 263083395960, 1093683448440, 4535210472000, 18764563053600, 77485731403080, 319402222692696, 1314511549519104
Offset: 3

Views

Author

Theodore M. Mishura, May 12 2015

Keywords

Links

Crossrefs

Cf. A002055, A114593.

Programs

  • Mathematica
    a[n_] := 2 Sum[Sum[
   Binomial[j + k, k]*Binomial[2 n - k - j - 1, n - k + 1], {k,
    n}], {j, 0, n - 2}]
CoefficientList[Series[(2(-1+Sqrt[1-4x]+x(7-5Sqrt[1-4x] +2x(-6+2Sqrt[ 1-4x] +x))))/ ((1-4x)^(3/2)x^2), {x, 0, 20}],x] (* Benedict W. J. Irwin, Jul 13 2016 *)
  • PARI
    a(n) = 2*n*binomial(2*n,n-2) \\ Charles R Greathouse IV, May 21 2015

Formula

a(n) = 2*(n-1)*binomial(2*n-2, n-3).
a(n) = A114593(n^2).
a(n) = 4*A002055(n+3). - Alois P. Heinz, May 21 2015
From Benedict W. J. Irwin, Jul 13 2016: (Start)
G.f.: 2*(s-1+x*(7-5*s+2*x*(2*s-6+x)))/(s^3x^2), where s=sqrt(1-4*x).
E.g.f: 2*E^(2*x)*x*(BesselI(1,2*x)+2*BesselI(2,2*x)+BesselI(3,2*x)).
(End)

Extensions

Terms a(22) and beyond from Andrew Howroyd, Nov 05 2019

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

Content is available under The OEIS End-User License Agreement.