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A108437 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 the first peak equal to k. 0
1, 1, 2, 5, 2, 1, 10, 28, 13, 11, 3, 1, 66, 196, 90, 89, 34, 18, 4, 1, 498, 1532, 694, 736, 311, 197, 66, 26, 5, 1, 4066, 12804, 5738, 6344, 2800, 1937, 762, 367, 110, 35, 6, 1, 34970, 111964, 49758, 56576, 25560, 18636, 7953, 4263, 1551, 615, 167, 45, 7, 1 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,3

COMMENTS

Row n contains 2n terms. Row sums yield A027307. T(n,1)=A027307(n-1).

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.=G=G(t, z)=1/(1-tzA-t^2*zA^2)-1, 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)=2 because we have uUddd and Uuddd.

Triangle begins:

1,1;

2,5,2,1;

10,28,13,11,3,1;

66,196,90,89,34,18,4,1;

MAPLE

A:=(2/3)*sqrt((z+3)/z)*sin((1/3)*arcsin(sqrt(z)*(z+18)/(z+3)^(3/2)))-1/3: G:=1/(1-t*z*A-t^2*z*A^2)-1: Gserz:=simplify(series(G, z=0, 10)): for n from 1 to 9 do P[n]:=sort(coeff(Gserz, z^n)) od: for n from 1 to 9 do seq(coeff(P[n], t^k), k=1..2*n) od; # yields sequence in triangular form

CROSSREFS

Cf. A027307.

Sequence in context: A061176 A180957 A124780 * A226029 A152765 A327867

Adjacent sequences:  A108434 A108435 A108436 * A108438 A108439 A108440

KEYWORD

nonn,tabf

AUTHOR

Emeric Deutsch, Jun 04 2005

STATUS

approved

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Last modified October 18 05:26 EDT 2019. Contains 328146 sequences. (Running on oeis4.)