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

Year-end appeal: Please make a donation to the OEIS Foundation to support ongoing development and maintenance of the OEIS. We are now in our 56th year, we are closing in on 350,000 sequences, and we’ve crossed 9,700 citations (which often say “discovered thanks to the OEIS”).

 Hints (Greetings from The On-Line Encyclopedia of Integer Sequences!)
 A114499 Triangle read by rows: number of Dyck paths of semilength n having k 3-bridges of a given shape (0<=k<=floor(n/3)). A 3-bridge is a subpath of the form UUUDDD or UUDUDD starting at level 0. 2
 1, 1, 2, 4, 1, 12, 2, 37, 5, 119, 12, 1, 390, 36, 3, 1307, 114, 9, 4460, 376, 25, 1, 15452, 1262, 78, 4, 54207, 4310, 255, 14, 192170, 14934, 863, 44, 1, 687386, 52397, 2967, 145, 5, 2477810, 185780, 10338, 492, 20, 8992007, 664631, 36424, 1712, 70, 1 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,3 COMMENTS Row n has 1+floor(n/3) terms. Row sums are the Catalan numbers (A000108). Column 0 is A114500. Sum(kT(n,k),k=0..floor(n/3))=Catalan(n-2) (n>=3; A000108). LINKS FORMULA G.f.=1/(1+z^3-tz^3-zC), where C=[1-sqrt(1-4z)]/(2z) is the Catalan function. EXAMPLE T(4,1)=2 because we have UD(UUUDDD) and (UUUDDD)UD (or UD(UUDUDD) and (UUDUDD)UD). The 3-bridges are shown between parentheses. Triangle starts: 1; 1; 2; 4,1; 12,2; 37,5; 119,12,1; 390,36,3; 1307,114,9; MAPLE C:=(1-sqrt(1-4*z))/2/z: G:=1/(1-z*C+z^3-t*z^3): Gser:=simplify(series(G, z=0, 20)): P[0]:=1: for n from 1 to 17 do P[n]:=coeff(Gser, z^n) od: for n from 0 to 17 do seq(coeff(t*P[n], t^j), j=1..1+floor(n/3)) od; # yields sequence in triangular form CROSSREFS Cf. A000108, A114500. Sequence in context: A246188 A135333 A124503 * A030730 A117131 A204117 Adjacent sequences:  A114496 A114497 A114498 * A114500 A114501 A114502 KEYWORD nonn,tabf AUTHOR Emeric Deutsch, Dec 04 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 | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

Last modified December 2 05:01 EST 2021. Contains 349437 sequences. (Running on oeis4.)