0,2

Partial sums of the Motzkin sequence (A001006). - Emeric Deutsch (deutsch(AT)duke.poly.edu), Jul 12 2004

a(n) = number of distinct ordered trees obtained by branch-reducing the ordered trees on n+1 edges. - David Callan (callan(AT)stat.wisc.edu), Oct 24 2004

a(n)= the number of consecutive horizontal steps at height 0 of all Motzkin paths from (0,0) to (n,0) starting with a horizontal step. - Charles Moore (chamoore(AT)howard.edu), Apr 15 2007

Equals row sums of triangle A136788 - Gary W. Adamson (qntmpkt(AT)yahoo.com), Jan 21 2008

G.f.: A(x) = 1/(1-x)^2 + x^2*A(x)^2.

a(n)=sum{k=0..floor((n+1)/2), binomial(n+1, 2k+1)binomial(2k, k)/(k+1)} - Paul Barry (pbarry(AT)wit.ie), Nov 29 2004

a(n) = n + 1 + sum_k a(k-1)a(n-k-1), starting from a(n)=0 for n negative. - Henry Bottomley (se16(AT)btinternet.com), Feb 22 2005

a(n)=sum{k=0..n, sum{j=0..n-k, C(j)C(n-k, 2j)}}; - Paul Barry (pbarry(AT)wit.ie), Aug 19 2005

G.f.: c(x^2/(1-x)^2)/(1-x)^2, c(x) the g.f. of A000108; a(n)=sum{k=0..floor(n/2), C(n+1,n-2k)C(k)}; - Paul Barry (pbarry(AT)wit.ie), May 31 2006

Binomial transform of doubled Catalan sequence 1,1,1,1,2,2,5,5,14,14,... - Paul Barry (pbarry(AT)wit.ie), Nov 17 2005

Row sums of Pascal-Catalan triangle A086617. - Paul Barry (pbarry(AT)wit.ie), Nov 17 2005

g(z)=(1-z-sqrt(1-2z-3z^2))/2z-2z^2 - Charles Moore (chamoore(AT)howard.edu), Apr 15 2007

a(0)=1, a(1)=2, a(2)=3+1=4, a(3)=4+4=8, a(4)=5+10+2=17, a(5)=6+20+12=38, are upward antidiagonal sums of triangle A086614:

{1},

{2,1},

{3,4,2},

{4,10,12,5},

{5,20,42,40,14},

{6,35,112,180,140,42}, ...

For example with n=2, the 5 ordered trees (A000108) on 3 edges are

|...|..../\.../\.../|\..

|../.\..|......|........

|.......................

Suppressing nonroot vertices of outdegree 1 (branch-reducing) yields

|...|..../\.../\../|\..

.../.\.................

of which 4 are distinct. So a(2)=4.

a(4)=8 because we have HHHH, HHUD, HUDH, HUHD

Cf. A086614 (triangle), A086616 (row sums).

Cf. A001006.

Cf. A136788.

Sequence in context: A025276 A006461 A003007 this_sequence A081124 A090901 A101516

Adjacent sequences: A086612 A086613 A086614 this_sequence A086616 A086617 A086618

nonn

Paul D. Hanna (pauldhanna(AT)juno.com), Jul 24 2003

Edited by N. J. A. Sloane (njas(AT)research.att.com), Oct 16 2006