The OEIS mourns the passing of Jim Simons and is grateful to the Simons Foundation for its support of research in many branches of science, including the OEIS.
The OEIS is supported by the many generous donors to the OEIS Foundation.

 Hints (Greetings from The On-Line Encyclopedia of Integer Sequences!)
 A109158 Triangle read by rows: T(n,k) is number of paths from (0,0) to (3n,0) that stay in the first quadrant (but may touch the horizontal axis), consisting of steps u=(2,1),U=(1,2), or d=(1,-1) and having height of last peak equal to k. 0
 1, 1, 2, 4, 3, 1, 10, 20, 18, 12, 5, 1, 66, 132, 122, 92, 54, 24, 7, 1, 498, 996, 930, 732, 478, 264, 118, 40, 9, 1, 4066, 8132, 7634, 6140, 4214, 2552, 1342, 600, 218, 60, 11, 1, 34970, 69940, 65874, 53676, 37910, 24136, 13782, 7016, 3122, 1180, 362, 84, 13, 1 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,3 COMMENTS Row n has 2n terms. Row sums yield A027307. T(n,1)=A027307(n-1). T(n,2)=2*A027307(n-1) for n>=2. LINKS Table of n, a(n) for n=1..56. Emeric Deutsch, Problem 10658: Another Type of Lattice Path, American Math. Monthly, 107, 2000, 368-370. FORMULA G.f.=tz(1+t)/[1-tz-t^2z-(1+t)zA-zA^2], where A=1+zA^2+zA^3=(2/3)*sqrt((z+3)/z)*sin((1/3)*arcsin(sqrt(z)*(z+18)/(z+3)^(3/2)))-1/3 (the g.f. of A027307). EXAMPLE T(2,3)=3 because we have uUddd, UdUddd and Uuddd. Triangle begins: 1,1; 2,4,3,1; 10,20,18,12,5,1; 66,132,122,92,54,24,7,1; MAPLE A:=(2/3)*sqrt((z+3)/z)*sin((1/3)*arcsin(sqrt(z)*(z+18)/(z+3)^(3/2)))-1/3: G:=t*z*(1+t)/(1-t*z-t^2*z-(1+t)*z*A-z*A^2): Gser:=simplify(series(G, z=0, 10)): for n from 1 to 8 do P[n]:=coeff(Gser, z^n) od: for n from 1 to 8 do seq(coeff(P[n], t^k), k=1..2*n) od; # yields sequence in triangular form CROSSREFS Cf. A027307. Sequence in context: A274329 A322398 A341606 * A307500 A049245 A123547 Adjacent sequences: A109155 A109156 A109157 * A109159 A109160 A109161 KEYWORD nonn,tabf AUTHOR Emeric Deutsch, Jun 21 2005 STATUS approved

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

Last modified May 19 00:35 EDT 2024. Contains 372666 sequences. (Running on oeis4.)