0,5

Row-sum of signed Catalan triangle A009766. (Mma program and comment from wouter.meeussen(AT)pandora.be.)

There are two schools of thought about the best indexing for these numbers. Deutsch and Shapiro have a(4) = 6 whereas here a(5) = 6. The formulae given here use both labelings.

Comments from Douglas Rogers, Oct 18 2005:

"I notice that you have some other zero-one evaluations of binary bracketings (such as A055395). But if you have an operation # with 0#0 = 1#0 = 1, 0#1 = 1#1 = 0,

and look at the number of bracketings of a string of n 0's that come out 0, you get another instance of the Fine numbers.

For Z = 1 + x(ZW + WW) = 1 + x CW and W = x(ZZ + ZW) = xZC. Hence Z = 1 + xxCCZ, the functional equational for the g.f. of the Fine numbers. Indeed, C = Z + W = Z + xCZ.

In terms of rooted planar trees with root of even degree, this says that of all rooted planar trees, some have root of even degree (Z) and some have root of odd degree (xCZ)."

Hankel transform of a(n+1)=[1,0,1,2,6,18,57,186,...] is A000012=[1,1,1,1,1,...] . - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Oct 24 2007

a(n+1) has g.f. 1/(1-0*x-x^2/(1-2*x-x^2/(1-2*x-x^2/(1-2*x-x^2/(..... (continued fraction). [From Paul Barry (pbarry(AT)wit.ie), Dec 02 2008]

Contribution from Gary W. Adamson (qntmpkt(AT)yahoo.com), Jan 09 2009: (Start)

Starting with offset 3 = iterates of M * [1,0,0,0,...] where M =

a tridiagonal matrix with [0,2,2,2,...] as the main diagonal and [1,1,1,...] as the super and subdiagonals. (End)

Starting with "1" and convolved with A068875 = the Catalan numbers with offset 1. [From Gary W. Adamson (qntmpkt(AT)yahoo.com), May 01 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).

Albert, M. H.; Aldred, R. E. L.; Atkinson, M. D.; Handley, C. C.; Holton, D. A.; and McCaughan, D. J.; Sorting classes, Electron. J. Combin. 12 (2005), no. 1, Research Paper 31, 25 pp.

Paul Barry, A Catalan Transform and Related Transformations on Integer Sequences, Journal of Integer Sequences, Vol. 8 (2005), Article 05.4.5.

P. J. Cameron, Some sequences of integers, Discrete Math., 75 (1989), 89-102; also in "Graph Theory and Combinatorics 1988", ed. B. Bollobas, Annals of Discrete Math., 43 (1989), 89-102.

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

E. Deutsch and L. Shapiro, Seventeen Catalan identities, Bull. Instit. Combin. Applic., 31 (2001), 31-38.

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

D. G. Rogers, Similarity relations on finite ordered sets, J. Combin. Theory, A 23 (1977), 88-98.

V. Strehl, A note on similarity relations, Discr. Math., 19 (1977), 99-102.

M. Aigner, Enumeration via ballot numbers, Discrete Math., 308 (2008), 2544-2563.

T. D. Noe, Table of n, a(n) for n = 0..200

E. Barcucci, E. Pergola, R. Pinzani and S. Rinaldi, ECO method and hill-free generalized Motzkin paths

N. T. Cameron, Random walks, trees and extensions of Riordan group techniques

J. W. Layman, The Hankel Transform and Some of its Properties, J. Integer Sequences, 4 (2001), #01.1.5.

P. Peart and W.-J. Woan, Generating Functions via Hankel and Stieltjes Matrices, J. Integer Seqs., Vol. 3 (2000), #00.2.1.

P. Peart and W.-J. Woan, Dyck Paths With No Peaks at Height k, J. Integer Sequences, 4 (2001), #01.1.3.

A. Robertson, D. Saracino and D. Zeilberger, Refined restricted permutations.

Index entries for sequences related to rooted trees

Catalan(n) = 2*a(n) + a(n-1), n >= 1.

G.f.: (1-sqrt(1-4*x))/(3-sqrt(1-4*x)) (compare g.f. for Catalan numbers, A000108) - from Emeric Deutsch (deutsch(AT)duke.poly.edu).

a(n) ~ 4^(n+1)/(9*n*sqrt(n*Pi)).

a(n)=(2/(n-1))sum((-2)^j*(j+1)binomial(2n-1, n-3-j), j=0..n-3), n>=2. - Emeric Deutsch (deutsch(AT)duke.poly.edu), Dec 26 2003

a(n) = 3*sum(binomial(2n-2j-2, n-1), j=0..floor((n-1)/2)) - binomial(2n, n) for n>0. - Emeric Deutsch (deutsch(AT)duke.poly.edu), Jan 28 2004

Reversion of g.f. (x-2x^2)/(1-x)^2. - R. Stephan, Mar 22 2004

a(n)=((-1)^n/2^n)*(-3/4-(1/4)*sum{k=0..n, C(1/2, k)8^k})+0^n; a(n)=((-1)^n/2^n)*(-3/4-(1/4)*sum{k=0..n, (-1)^(k-1)*2^k*(2k)!/((k!)^2*(2k-1))})+0^n. - Paul Barry (pbarry(AT)wit.ie), Jun 10 2005

Hankel determinant transform is 1-n. - Michael Somos Sep 17 2006

a(n+1)=A126093(n,0) . - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Mar 05 2007

Contribution from Paul Barry (pbarry(AT)wit.ie), Jan 17 2009: (Start)

G.f.: x*c(x)/(1+x*c(x)), c(x) the g.f. of A000108;

a(n+1)=sum{k=0..n, (-1)^k*C(2n-k,n-k)*(k+1)/(n+1)}. (End)

a(n) = 3*(-1/2)^(n+1) + GAMMA(n+1/2)*4^n*hypergeom([1, n+1/2],[n+2],-8)/(sqrt(Pi)*(n+1)!) (for n>0) [From Mark van Hoeij (hoeij(AT)math.fsu.edu), Nov 11 2009]

t1 := (1-sqrt(1-4*x))/(3-sqrt(1-4*x)); t2 := series(t1, x, 90); A000957 := n-> coeff(t2, x, n);

Table[ Plus@@Table[ (-1)^(m+n) (n+m)!/n!/m! (n-m+1)/(n+1), {m, 0, n} ], {n, 0, 36} ]

(PARI) {a(n)=if(n<1, 0, polcoeff( 1/(1+2/(1-sqrt(1-4*x+x*O(x^n)))), n))} /* Michael Somos Sep 17 2006 */

(PARI) {a(n)=if(n<1, 0, polcoeff( 1/(1+1/serreverse(x-x^2+x*O(x^n))), n))} /* Michael Somos Sep 30 2006 */

a(n)= (A064306(n-1)+(-1)^(n-1))/2^n, n >= 1. A column of A065600.

Sequence with signs: A064310. Bisections: A138413, A138414.

A068875 [From Gary W. Adamson (qntmpkt(AT)yahoo.com), May 01 2009]

Sequence in context: A064310 A126983 A104629 this_sequence A125305 A148458 A148459

Adjacent sequences: A000954 A000955 A000956 this_sequence A000958 A000959 A000960

nonn,nice,easy,new

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