1,3

a(n) = number of Dyck n-paths containing no peak at height 2 before the first return to ground level. Example: a(3)=3 counts UUUDDD, UDUUDD, UDUDUD. - David Callan (callan(AT)stat.wisc.edu), Jun 07 2006

Also number of order trees with n edges and having no even-length branches starting at the root. - Emeric Deutsch (deutsch(AT)duke.poly.edu), Mar 02 2007

Convolution of the Catalan sequence 1,1,2,5,14,42,... (A000108) and the Fine sequence 1,0,1,2,6,18,... (A000957). a(n)=A127541(n,0). - Emeric Deutsch (deutsch(AT)duke.poly.edu), Mar 02 2007

The Catalan transform of A008619. [From R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Nov 06 2008]

Hankel transform is F(2n+1). [From Paul Barry (pbarry(AT)wit.ie), Dec 01 2008]

Starting with offset 2 = iterates of M * [1,1,0,0,0,...] where M = a tridiagonal matrix with [0,2,2,2,...] in the main diagonal and [1,1,1,...] in the super and subdiagonals. [From Gary W. Adamson (qntmpkt(AT)yahoo.com), Jan 09 2009]

Equals INVERT transform of A032357 [From Gary W. Adamson (qntmpkt(AT)yahoo.com), Apr 10 2009]

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

E. Deutsch and L. Shapiro, A survey of the Fine numbers, Discrete Math., 241 (2001), 241-265.

Fine, Terrence; Extrapolation when very little is known about the source. Information and Control 16 (1970), 331-359.

Index entries for sequences related to rooted trees

Equals A000957(n) + A000957(n+1).

G.f.:=(1-x-(1+x)*sqrt(1-4*x))/(2x(x+2)); - Paul Barry (pbarry(AT)wit.ie), Jan 26 2007

G.f.=zC/(1-z^2*C^2), where C=(1-sqrt(1-4z))/(2z) is the Catalan function. - Emeric Deutsch (deutsch(AT)duke.poly.edu), Mar 02 2007

a(n+1)=Sum_{k, 0<=k<=[n/2]}A039599(n-k,k). - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Mar 13 2007

a(n) = (-1/2)^n*(-2-5*sum((-8)^k*GAMMA(1/2+k)*(4/5+k)/(sqrt(Pi)*GAMMA(k+3)),k = 1 .. -1+n)) [From Mark van Hoeij (hoeij(AT)math.fsu.edu), Nov 11 2009]

a(n)+a(n+1)=A135339(n+1). [From Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Dec 02 2009]

g:=(1-x-(1+x)*sqrt(1-4*x))/2/x/(x+2): gser:=series(g, x=0, 30): seq(coeff(gser, x, n), n=1..26); - Emeric Deutsch (deutsch(AT)duke.poly.edu), Mar 02 2007

A column of A065602.

Cf. A127541, A127539, A000108, A000957.

A032357 [From Gary W. Adamson (qntmpkt(AT)yahoo.com), Apr 10 2009]

Sequence in context: A006365 A046919 A046342 this_sequence A148782 A148783 A084205

Adjacent sequences: A000955 A000956 A000957 this_sequence A000959 A000960 A000961

nonn,easy,new

N. J. A. Sloane (njas(AT)research.att.com).