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 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-François 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

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