0,3

This sequence represents the expected saturation of a binary search tree (or BST) on n nodes times the number of binary search trees on n nodes, or alternatively, the sum of the saturation of all binary search trees on n nodes. - Marko Riedel (mriedel(AT)neuearbeit.de), Jan 24 2002

1->12, 2->123, 3->1234 etc. starting with 1, gives A007001: 1, 12, 12123, 12123121231234... suming the digits gives this sequence. - Miklos Kristof (kristmikl(AT)freemail.hu), Nov 05 2002

a(n-1) = number of n-th generation vertices in the tree of sequences with unit increase labeled by 2 (cf. Zoran Sunik reference.) - Benoit Cloitre (benoit7848c(AT)orange.fr), Oct 07 2003

With offset 1, number of permutations beginning with 12 and avoiding 32-1.

Number of lattice paths from (0,0) to (n,n) with steps E=(1,0) and N=(0,1) which touch but do not cross the line x-y=1. - Herbert Kociemba (kociemba(AT)t-online.de), May 24 2004

a(n)=number of Dyck (n+1)-paths that start with UU. For example, a(2)=3 counts UUUDDD, UUDUDD, UUDDUD. - David Callan (callan(AT)stat.wisc.edu), Dec 08 2004

Hankel transform is (0,-1,-1,0,1,1,0,-1,-1, 0...). Hankel transform of a(n+1) is (1,0,-1,-1,0,1,1,0,-1,-1,0...). - Paul Barry (pbarry(AT)wit.ie), Feb 08 2008

Starting with offset 1 = row sums of triangle A154558 [From Gary W. Adamson (qntmpkt(AT)yahoo.com), Jan 11 2009]

Starting with offset 1 equals INVERT transform of A014137, partial sums of the Catalan nubmers: (1, 2, 4, 9, 23,...). [From Gary W. Adamson (qntmpkt(AT)yahoo.com), May 15 2009]

Contribution from Aoife Hennessy (aoife.hennessy(AT)gmail.com), Sep 07 2009: (Start)

With offset 1, a(n) is the binomial transform of the shortened

Motzkin numbers: 1, 2, 4, 9, 21, 51, 127, 323....(A001006) (End)

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).

N. S. S. Gu, N. Y. Li and T. Mansour, 2-Binary trees: bijections and related issues, Discr. Math., 308 (2008), 1209-1221.

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

S. J. Cyvin and I. Gutman, Kekule structures in benzenoid hydrocarbons, Lecture Notes in Chemistry, No. 46, Springer, New York, 1988 (see p. 196).

V. E. Hoggatt, Jr. and M. Bicknell, Catalan and related sequences arising from inverses of Pascal's triangle matrices, Fib. Quart., 14 (1976), 395-405.

A. Papoulis, A new method of inversion of the Laplace transform, Quart. Applied Math. 14 (1956), 405ff.

J. Riordan, The distribution of crossings of chords joining pairs of 2n points on a circle, Math. Comp., 29 (1975), 215-222.

Zoran Sunik, Self describing sequences and the Catalan family tree, Elect. J. Combin., 10 (No. 1, 2003).

T. D. Noe, Table of n, a(n) for n=0..100

D. Callan, A recursive bijective approach to counting permutations...

R. K. Guy, Catwalks, Sandsteps and Pascal Pyramids, J. Integer Seqs., Vol. 3 (2000), #00.1.6

S. Kitaev and T. Mansour, Simultaneous avoidance of generalized patterns.

S. Kitaev, Generalized pattern avoidance with additional restrictions, Sem. Lothar. Combinat. B48e (2003).

Comment from wolfdieter.lang(AT)physik.uni-karlsruhe.de: G.f.: x*(c(x))^3 = (-1+(1-x)*c(x))/x, c(x) = g.f. for Catalan numbers. Also a(n)=3*n*Catalan(n)/(n+2).

For n>1, a(n)=3a(n-1)+Sum[a(k)a(n-k-2), k=1,...,n-3]. - John W. Layman (layman(AT)math.vt.edu), Dec 13 2002; proved by Michael Somos, Jul 05, 2003

G.f. is A(x) = C(x)(1-x)/x-1/x = x(1+xC(x)^2)C(x)^2 where C(x) is g.f. for Catalan numbers, A000108.

G.f. satisfies x^2A(x)^2+(3x-1)A(x)+x=0.

Series reversion of g.f. A(x) is -A(-x). - Michael Somos, Jan 21 2004

a(n+1)=sum(i+j+k=n, C(i)C(j)C(k)) with i, j, k>=0 and where C(k) denotes the k-th Catalan number. - Benoit Cloitre, Nov 09 2003

An inverse Chebyshev transform of x^2. - Paul Barry (pbarry(AT)wit.ie), Oct 13 2004

The sequence is 0, 0, 1, 0, 3, 0, 9, 0, ...with zeros restored. Second binomial transform of (-1)^n*A005322(n). The g.f. is transformed to x^2 under the Chebyshev transformation A(x)->(1/(1+x^2))A(x/(1+x^2)). For a sequence b(n), this corresponds to taking sum{k=0..floor(n/2), C(n-k, k)(-1)^k*b(n-2k)}, or sum{k=0..n, C((n+k)/2, k)b(k)(-1)^((n-k)/2)(1+(-1)^(n-k))/2}. - Paul Barry (pbarry(AT)wit.ie), Oct 13 2004

G.f.: (c(x^2)(1-x^2)-1)/x^2, c(x) the g.f. of A000108; a(n)=sum{k=0..n, (k+1)C(n, (n-k)/2)(-1)^k(C(2, k)-2C(1, k)+C(0, k))(1+(-1)^(n-k))/(n+k+2)} - Paul Barry (pbarry(AT)wit.ie), Oct 13 2004

C(n+1)-C(n)=sum{k=0..n, C(n,k)*2^(n-k)*(-1)^(k+1)*C(k,floor((k-1)/2))}; - Paul Barry (pbarry(AT)wit.ie), Feb 16 2006

E.g.f.: exp(2x)(Bessel_I(1,2x)-Bessel_I(2,2x)); - Paul Barry (pbarry(AT)wit.ie), Jun 04 2007

a(n)=(1/pi)*int(x^n*(x-1)*sqrt(x(4-x))/(2x),x,0,4); - Paul Barry (pbarry(AT)wit.ie), Feb 08 2008

A000245 := n -> 3*binomial(2*n, n-1)/(n+2);

(PARI) a(n)=if(n<1, 0, 3*(2*n)!/(n+2)!/(n-1)!)

(Other) sage: [catalan_number(i+1)-catalan_number(i) for i in xrange(0, 28)]# [From Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), May 17 2009]

First differences of Catalan numbers A000108.

T(n, n+3) for n=0, 1, 2, ..., array T as in A047072. Also a diagonal of A059365 and of A009766.

Cf. A099364.

A diagonal of any of the essentially equivalent arrays A009766, A030237, A033184, A059365, A099039, A106566, A130020, A047072.

Cf. A000108 A000245 A002057 A000344 A003517 A000588 A003518 A003519 A001392.

A154558 [From Gary W. Adamson (qntmpkt(AT)yahoo.com), Jan 11 2009]

Cf. A014137 [From Gary W. Adamson (qntmpkt(AT)yahoo.com), May 15 2009]

Sequence in context: A094826 A033190 A071724 this_sequence A143739 A047047 A071744

Adjacent sequences: A000242 A000243 A000244 this_sequence A000246 A000247 A000248

nonn,easy,nice

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

I changed the description and added an initial 0, to make this coincide with the first differences of the Catalan numbers A000108. Some of the other lines will need to be changed as a result. - N. J. A. Sloane (njas(AT)research.att.com), Oct 31, 2003.