|
Search: id:A097861
|
|
|
| A097861 |
|
Number of humps in all Motzkin paths of length n. (A hump is an upstep followed by 0 or more flatsteps followed by a downstep.). |
|
+0
1
|
|
| 0, 1, 3, 9, 25, 70, 196, 553, 1569, 4476, 12826, 36894, 106470, 308113, 893803, 2598313, 7567465, 22076404, 64498426, 188689684, 552675364, 1620567763, 4756614061, 13974168190, 41088418150, 120906613075, 356035078101
(list; graph; listen)
|
|
|
OFFSET
|
1,3
|
|
|
FORMULA
|
G.f.=[1-z-sqrt(1-2z-3z^2)]/[2(1-z)sqrt(1-2z-3z^2)].
a(n) = (A002426(n)-1)/2. E.g.f.: exp(x)*(BesselI(0, 2*x)-1)/2. - Vladeta Jovovic (vladeta(AT)Eunet.yu), Jul 24 2005
sum (binomial(n,j)*binomial(n-j,j)/2,j=1..n). - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Sep 24 2006
|
|
EXAMPLE
|
a(3)=3 because in all Motzkin paths of length 3 we have 3 humps, shown between parentheses: FFF, F(UD), (UD)F, (UFD) (here U=(1,1), F=(1,0), D=(1,-1)).
|
|
MAPLE
|
G:=(1-z-sqrt(1-2*z-3*z^2))/2/(1-z)/sqrt(1-2*z-3*z^2): Gser:=series(G, z=0, 33): seq(coeff(Gser, z^n), n=1..32);
a:=n->sum (binomial(n, j)*binomial(n-j, j)/2, j=1..n): seq(a(n), n=1..27); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Sep 24 2006
|
|
CROSSREFS
|
Cf. A097229.
Sequence in context: A103780 A098182 A007046 this_sequence A058719 A046661 A101197
Adjacent sequences: A097858 A097859 A097860 this_sequence A097862 A097863 A097864
|
|
KEYWORD
|
nonn
|
|
AUTHOR
|
Emeric Deutsch (deutsch(AT)duke.poly.edu), Sep 01 2004
|
|
|
Search completed in 0.002 seconds
|
