login
The OEIS Foundation is supported by donations from users of the OEIS and by a grant from the Simons Foundation.

 

Logo


Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A293946 a(n) = number of lattice paths from (0,0) to (3n,2n) which lie wholly below the line 3y=2x, only touching at the endpoints. 5
1, 2, 19, 293, 5452, 112227, 2460954, 56356938, 1332055265, 32251721089, 795815587214, 19939653287183, 505943824579282, 12974266405435153, 335717028959470883, 8754495459668971998, 229836484204401559180, 6069875377376291350173, 161145418968823760038557 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

LINKS

Robert Israel, Table of n, a(n) for n = 0..687 (corrected by Ray Chandler, Jan 19 2019)

M. T. L. Bizley, Derivation of a new formula for the number of minimal lattice paths from (0, 0) to (km, kn) having just t contacts with the line my = nx and having no points above this line; and a proof of Grossman's formula for the number of paths which may touch but do not rise above this line, Journal of the Institute of Actuaries, Vol. 80, No. 1 (1954): 55-62.[Cached copy; Annotated copy of page 59]

Bryan Ek, Lattice Walk Enumeration, arXiv:1803.10920 [math.CO], 2018.

Bryan Ek, Unimodal Polynomials and Lattice Walk Enumeration with Experimental Mathematics, arXiv:1804.05933 [math.CO], 2018.

FORMULA

a(n) = T(3n,2n) where T is the triangle from A294207. - Danny Rorabaugh, Oct 24 2017

G.f. A(z) satisfies A^10-19*A^9+162*A^8-816*A^7+2688*A^6+(-2*z-6048)*A^5+(19*z+9408)*A^4+(-73*z-9984)*A^3+(142*z+6912)*A^2+(-140*z-2816)*A+z^2+56*z+512=0 (Proven). - Bryan T. Ek, Oct 30 2017

MAPLE

f:= proc(n) local U, x, y;

  U:= Array(1..3*n, 0..2*n);

  U[3*n, 2*n]:= 1:

  for x from 3*n to 1 by -1 do

    for y from ceil(2/3*x)-1 to 0 by -1 do

      if x+1 <= 3*n then U[x, y]:= U[x+1, y] fi;

      if y+1 < 2/3*x or x=3*n then U[x, y]:= U[x, y]+U[x, y+1] fi;

  od od:

  U[1, 0];

end proc:

map(f, [$1..30]); # Robert Israel, Oct 24 2017

MATHEMATICA

T[_, 0] = 1; T[n_, k_] := T[n, k] = Which[0 < k < 2(n-1)/3, T[n-1, k] + T[n, k-1], 2(n-1) <= 3k <= 2n, T[n, k-1]];

a[n_] := T[3n, 2n];

Table[a[n], {n, 0, 30}] (* Jean-Fran├žois Alcover, Jul 10 2018, after Danny Rorabaugh *)

CROSSREFS

Cf. A000108, A060941, A322634.

Sequence in context: A191806 A252710 A065923 * A094476 A304637 A119773

Adjacent sequences:  A293943 A293944 A293945 * A293947 A293948 A293949

KEYWORD

nonn,easy

AUTHOR

N. J. A. Sloane, Oct 24 2017

EXTENSIONS

More terms from Robert Israel, Oct 24 2017

Offset changed and a(0) by Danny Rorabaugh, Oct 24 2017

STATUS

approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified July 8 01:42 EDT 2020. Contains 335502 sequences. (Running on oeis4.)