1,2

Also number of lattice paths from (0,0) to (n,n), with steps (1,0),(0,1) and (1,1), that never rise above the line y=x and the steps (1,1) are colored red or blue. - Emeric Deutsch (deutsch(AT)duke.poly.edu), May 28 2003

The Hankel transform (see A001906 for definition) of this sequence forms A049656(n+1)= [1, 3, 27, 729, 59049, 14348907, ... ] . - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Aug 29 2006

With a(0)=0, this is the series reversion of x(1-x)/(1+2x). [From Paul Barry (pbarry(AT)wit.ie), Oct 18 2009]

Paul Barry, On Integer-Sequence-Based Constructions of Generalized Pascal Triangles, Journal of Integer Sequences, Vol. 9 (2006), Article 06.2.4.

Index entries for sequences related to rooted trees

(1-2z-sqrt(4z^2-8z+1))/2z.

For n>0, a(n)=(1/n)*sum(k=0, n, 3^k*C(n, k)*C(n, k-1)) - Benoit Cloitre (benoit7848c(AT)orange.fr), May 10 2003

a(1)=1, a(n)=2*a(n-1)+sum(i=1, n-1, a(i)*a(n-i)). - Benoit Cloitre (benoit7848c(AT)orange.fr), Mar 16 2004

The Hankel transform (see A001906 for definition) of this sequence form A049656(n+1)= [1, 3, 27, 729, 59049, 14348907, ... ] . - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Aug 29 2006

2*a(n)=A054872(n+1). - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Aug 17 2007

Contribution from Paul Barry (pbarry(AT)wit.ie), Feb 01 2009: (Start)

G.f.: 1/(1-2x-x/(1-2x-x/(1-2x-x/(1-2x-x/(1-... (continued fraction);

a(n)=sum{k=0..n, C(n+k,2k)*2^(n-k)*A000108(k)}. (End)

G.f.: 1/(1-3x/(1-x/(1-3x/(1-x/(1-3x/(1-x/(1-3x/(1-... (continued fraction). [From Paul Barry (pbarry(AT)wit.ie), Oct 18 2009]

(PARI) a(n)=if(n<1, 1, sum(k=0, n, 3^k*binomial(n, k)*binomial(n, k-1))/n)

Essentially the same as A025231.

Cf. A006318.

Sequence in context: A101106 A165310 A133158 this_sequence A151498 A103370 A094149

Adjacent sequences: A047888 A047889 A047890 this_sequence A047892 A047893 A047894

nonn,eigen,easy

Louis Shapiro (lshapiro(AT)howard.edu)

More terms from Christian Bower, Dec 11 1999