login
This site is supported by donations to The OEIS Foundation.

 

Logo

Please make a donation to keep the OEIS running. We are now in our 55th year. In the past year we added 12000 new sequences and reached 8000 citations (which often say "discovered thanks to the OEIS"). We need to raise money to hire someone to manage submissions, which would reduce the load on our editors and speed up editing.
Other ways to donate

Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A097612 Triangle read by rows: T(n,k) is the number of Motzkin paths of length n having k returns (i.e. down steps hitting the x-axis). 0
1, 1, 1, 1, 1, 3, 1, 7, 1, 1, 15, 5, 1, 32, 17, 1, 1, 70, 49, 7, 1, 159, 131, 31, 1, 1, 375, 339, 111, 9, 1, 914, 869, 354, 49, 1, 1, 2288, 2233, 1056, 209, 11, 1, 5850, 5784, 3031, 773, 71, 1, 1, 15210, 15132, 8515, 2613, 351, 13, 1, 40081, 39990, 23659, 8329, 1476, 97, 1 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,6

COMMENTS

Row sums are the Motzkin numbers (A001006).

LINKS

Table of n, a(n) for n=0..63.

FORMULA

G.f.= 2/[2-2z-t+tz+tsqrt(1-2z-3z^2)].

EXAMPLE

Triangle begins:

1;

1;

1,1;

1,3;

1,7,1;

1,15,5;

Row n has 1+floor(n/2) terms.

T(5,2)=5 because HU(D)U(D), U(D)HU(D), U(D)U(D)H, U(D)UH(D) and UH(D)U(D) are the only Motzkin paths of length 5 with 2 returns (shown between parentheses); here U=(1,1), H=(1,0) and D=(1,-1).

MAPLE

G:= 2/(2-2*z-t+t*z+t*sqrt(1-2*z-3*z^2)) : Gser:=simplify(series(G, z=0, 16)): P[0]:=1: for n from 1 to 15 do P[n]:=sort(coeff(Gser, z^n)) od: seq(seq(coeff(t*P[n], t^k), k=1..1+floor(n/2)), n=0..15);

CROSSREFS

Cf. A001006.

Sequence in context: A257597 A097229 A097862 * A136011 A227984 A021991

Adjacent sequences:  A097609 A097610 A097611 * A097613 A097614 A097615

KEYWORD

nonn,tabf

AUTHOR

Emeric Deutsch, Aug 30 2004

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 December 14 01:33 EST 2019. Contains 329978 sequences. (Running on oeis4.)