|
Search: id:A101895
|
|
|
| A101895 |
|
Triangle read by rows: T(n,k) is the number of Schroeder paths of length 2n and having k peaks at even height. |
|
+0
2
|
|
| 2, 5, 1, 15, 6, 1, 51, 30, 8, 1, 188, 144, 51, 10, 1, 731, 685, 300, 77, 12, 1, 2950, 3258, 1695, 532, 108, 14, 1, 12235, 15533, 9348, 3455, 854, 144, 16, 1, 51822, 74280, 50729, 21538, 6245, 1280, 185, 18, 1, 223191, 356283, 272128, 130375, 43278, 10387, 1824
(list; table; graph; listen)
|
|
|
OFFSET
|
1,1
|
|
|
COMMENT
|
A Schroeder path of length 2n is a lattice path starting from (0,0), ending at (2n,0), consisting only of steps U=(1,1) (up steps), D=(1,-1) (down steps) and H=(2,0) (level steps) and never going below the x-axis. Schroeder paths are counted by the large Schroeder numbers (A006318). Row sums are the large Schroeder numbers (A006318). Column 0 yields A007317. Column 1 yields A026376.
|
|
FORMULA
|
G.f.=G=G(t, z) satisfies z(1-z)G^2-(1-z)(1-tz)G+1-tz=0.
|
|
EXAMPLE
|
T(3,1)=6 because we have HU(UD)D, U(UD)DH, UH(UD)D, U(UD)HD, UDU(UD)D, and
U(UD)DUD, the peaks at even height being shown between parentheses.
Triangle begins:
2;
5,1;
15,6,1;
51,30,8,1;
188,144,51,10,1;
|
|
MAPLE
|
G := 1/2/(-z+z^2)*(-1+t*z+z-t*z^2+sqrt(1-2*t*z-6*z+8*t*z^2+t^2*z^2-2*t^2*z^3+5*z^2-6*t*z^3+t^2*z^4)): Gser:=simplify(series(G, z=0, 14)): for n from 1 to 12 do P[n]:=coeff(Gser, z^n) od: for n from 1 to 12 do seq(coeff(t*P[n], t^k), k=1..n) od; # yields the sequence in triangular form
|
|
CROSSREFS
|
Cf. A006318, A007317, A026376, A101894.
Sequence in context: A118919 A101282 A145879 this_sequence A110220 A119518 A111797
Adjacent sequences: A101892 A101893 A101894 this_sequence A101896 A101897 A101898
|
|
KEYWORD
|
nonn,tabl
|
|
AUTHOR
|
Emeric Deutsch (deutsch(AT)duke.poly.edu), Dec 20 2004
|
|
|
Search completed in 0.002 seconds
|
