0,4

Essentially the same as A025246.

Number of lattice paths in the first quadrant from (0,0) to (n,0) using only steps H=(1,0), U=(1,1) and D=(2,-1). E.g. a(5)=7 because we have HHHHH, HHUD, HUDH, HUHD, UDHH, UHDH and UHHD. - Emeric Deutsch (deutsch(AT)duke.poly.edu), Dec 25 2003

Also number of peakless Motzkin paths of length n with no double rises; in other words, Motzkin paths of length n with no UD's and no UU's, where U=(1,1) and D=(1,-1). E.g. a(5)=7 because we have HHHHH, HHUHD, HUHDH, HUHHD, UHDHH, UHHDH and UHHHD, where H=(1,0). - Emeric Deutsch (deutsch(AT)duke.poly.edu), Jan 09 2004

Series reversion of g.f. A(x) is -A(-x) (if offset 1). - Michael Somos, Jul 13 2003

Hankel transform is A010892(n+1). [From Paul Barry (pbarry(AT)wit.ie), Sep 19 2008]

INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 666

G.f.: (1-x-sqrt((1-x)^2-4x^3))/(2x^3)=A(x). y=xA(x) satisfies x-y+xy+(xy)^2=0.

a(n+1)=a(n)+a(0)a(n-2)+a(1)a(n-3)+...+a(n-2)a(0).

G.f.: (1/(1-x))c(x^3/(1-x)^2), c(x) the g.f. of A000108. [From Paul Barry (pbarry(AT)wit.ie), Sep 19 2008]

Contribution from Paul Barry (pbarry(AT)wit.ie), May 22 2009: (Start)

G.f.: 1/(1-x-x^3/(1-x-x^3/(1-x-x^3/(1-x-x^3/(1-... (continued fraction).

a(n)=sum{k=0..floor(n/3), C(n-k,2k)*A000108(k)}. (End)

Clear[ a ]; a[ 0 ]=1; a[ n_Integer ] := a[ n ]=a[ n-1 ]+Sum[ a[ k ]*a[ n-3-k ], {k, 0, n-3} ];

(PARI) a(n)=polcoeff((1-x-sqrt((1-x)^2-4*x^3+x^4*O(x^n)))/2, n+3)

Cf. A000108, A001006, A004148, A006318.

A025246(n+3)=a(n).

Sequence in context: A017995 A099155 A068031 this_sequence A025246 A112740 A136408

Adjacent sequences: A023428 A023429 A023430 this_sequence A023432 A023433 A023434

nonn,easy

Olivier Gerard (olivier.gerard(AT)gmail.com)