A000245 a(n) = 3*(2*n)!/((n+2)!*(n-1)!).

0, 1, 3, 9, 28, 90, 297, 1001, 3432, 11934, 41990, 149226, 534888, 1931540, 7020405, 25662825, 94287120, 347993910, 1289624490, 4796857230, 17902146600, 67016296620, 251577050010, 946844533674, 3572042254128, 13505406670700, 51166197843852, 194214400834356

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• easy
• nice

Comments

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, Jan 24 2002

1->12, 2->123, 3->1234 etc. starting with 1, gives A007001: 1, 12, 12123, 12123121231234... summing the digits gives this sequence. - Miklos Kristof, 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 Sunic reference). - Benoit Cloitre, 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, May 24 2004

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

a(n) is the number of Dyck (n+2)-paths that start with UUDU. For example, a(2)=3 counts UUDUDDUD, UUDUDUDD, UUDUUDDD. - David Scambler, Feb 14 2011

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, Feb 08 2008

Starting with offset 1 = row sums of triangle A154558. - Gary W. Adamson, Jan 11 2009

Starting with offset 1 equals INVERT transform of A014137, partial sums of the Catalan numbers: (1, 2, 4, 9, 23, ...). - Gary W. Adamson, May 15 2009

With offset 1, a(n) is the binomial transform of the shortened Motzkin numbers: 1, 2, 4, 9, 21, 51, 127, 323, ... (A001006). - Aoife Hennessy (aoife.hennessy(AT)gmail.com), Sep 07 2009

The Catalan sequence convolved with its shifted variant, e.g. a(5) = 90 = (1, 1, 2, 5, 14) dot (42, 14, 5, 2, 1) = (42 + 14 + 10 + 10 + 14 ) = 90. - Gary W. Adamson, Nov 22 2011

a(n+2) = A214292(2*n+3,n). - Reinhard Zumkeller, Jul 12 2012

With offset 3, a(n) is the number of permutations on {1,2,...,n} that are 123-avoiding, i.e., do not contain a three term monotone subsequence, for which the first ascent is at positions (3,4); see Connolly link. There it is shown in general that the k-th Catalan Convolution is the number of 123-avoiding permutations for which the first ascent is at (k, k+1). (For n=k, the first ascent is defined to be at positions (k,k+1) if the permutation is the decreasing permutation with no ascents.) - Anant Godbole, Jan 17 2014

With offset 3, a(n)=number of permutations on {1,2,...,n} that are 123-avoiding and for which the integer n is in the 3rd spot; see Connolly link. For example, there are 297 123-avoiding permutations on n=9 at which the element 9 is in the third spot. - Anant Godbole, Jan 17 2014

With offset 1, a(n) is the number of North-East paths from (0,0) to (n+1,n+1) that bounce off y = x to the right exactly once but do not cross y = x vertically. Details can be found in Section 4.4 in Pan and Remmel's link. - Ran Pan, Feb 01 2016

The total number of returns (downsteps which end on the line y=0) within the set of all Dyck paths from (0,0) to (2n,0). - Cheyne Homberger, Sep 05 2016

a(n) is the number of intervals of the form [s,w] that are distributive (equivalently, modular) lattices in the weak order on S_n, for a fixed simple reflection s. - Bridget Tenner, Jan 16 2020

a(n+2) is the number of coalescent histories for a pair (G,S) where G is a gene tree with 3-pseudocaterpillar shape and n leaves, S is a species tree with caterpillar shape and n leaves, and G and S have identical leaf labeling. - Noah A Rosenberg, Jun 15 2022

a(n) is the number of parking functions of size n avoiding the patterns 132, 213, and 312. - Lara Pudwell, Apr 10 2023

a(n) is the number of parking functions of size n avoiding the patterns 123 and 213. - Lara Pudwell, Apr 10 2023

References

• Pierre de la Harpe, Topics in Geometric Group Theory, Univ. Chicago Press, 2000, p. 11, coefficients of P_3(z).
• Ki Hang Kim, Douglas G. Rogers and Fred W. Roush, Similarity relations and semiorders. Proceedings of the Tenth Southeastern Conference on Combinatorics, Graph Theory and Computing (Florida Atlantic Univ., Boca Raton, Fla., 1979), pp. 577-594, Congress. Numer., XXIII-XXIV, Utilitas Math., Winnipeg, Man., 1979. MR0561081 (81i:05013)
• C. Krishnamachary and M. Bheemasena Rao, Determinants whose elements are Eulerian, prepared Bernoullian and other numbers, J. Indian Math. Soc., Vol. 14 (1922), pp. 55-62, 122-138 and 143-146.
• N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
• N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

• T. D. Noe, Table of n, a(n) for n = 0..100
• Marco Abrate, Stefano Barbero, Umberto Cerruti and Nadir Murru, Colored compositions, Invert operator and elegant compositions with the "black tie", Discrete Math. 335 (2014), 1-7. MR3248794
• Ayomikun Adeniran and Lara Pudwell, Pattern avoidance in parking functions, Enumer. Comb. Appl. 3:3 (2023), Article S2R17.
• Anwar Al Ghabra, K. Gopala Krishna, Patrick Labelle, and Vasilisa Shramchenko, Enumeration of multi-rooted plane trees, arXiv:2301.09765 [math.CO], 2023.
• F. S. Al-Kharousi, A. Umar, and M. M. Zubairu, On injective partial Catalan monoids, arXiv:2501.00285 [math.GR], 2024. See p. 2.
• Egor Alimpiev and Noah A Rosenberg, Enumeration of coalescent histories for caterpillar species trees and p-pseudocaterpillar gene trees, arXiv:2103.13464 [qbio.PE], 2021; Adv. Appl. Math. 131 (2021), 102265.
• Jean-Luc Baril, Classical sequences revisited with permutations avoiding dotted pattern, Electronic Journal of Combinatorics, 18 (2011), #P178.
• Jean-Luc Baril and Sergey Kirgizov, The pure descent statistic on permutations, Preprint, 2016.
• Jean-Luc Baril and Helmut Prodinger, Enumeration of partial Lukasiewicz paths, arXiv:2205.01383 [math.CO], 2022.
• Paul Barry, A Catalan Transform and Related Transformations on Integer Sequences, Journal of Integer Sequences, Vol. 8 (2005), Article 05.4.5.
• Paul Barry, Riordan Pseudo-Involutions, Continued Fractions and Somos 4 Sequences, arXiv:1807.05794 [math.CO], 2018.
• Paul Barry, On a Central Transform of Integer Sequences, arXiv:2004.04577 [math.CO], 2020.
• David Callan, A recursive bijective approach to counting permutations containing 3-letter patterns, arXiv:math/0211380 [math.CO], Nov 25 2002.
• Reinis Cirpons, James East, and James D. Mitchell, Transformation representations of diagram monoids, arXiv:2411.14693 [math.RA], 2024. See pp. 3, 33.
• S. Connolly, Z. Gabor and A. Godbole, The location of the first ascent in a 123-avoiding permutation, arXiv:1401.2691 [math.CO], 2014.
• S. J. Cyvin and I. Gutman, Kekulé structures in benzenoid hydrocarbons, Lecture Notes in Chemistry, No. 46, Springer, New York, 1988 (see p. 196).
• Olivier Danvy, Summa Summarum: Moessner's Theorem without Dynamic Programming, arXiv:2412.03127 [cs.DM], 2024. See p. 31.
• Dennis E. Davenport, Lara K. Pudwell, Louis W. Shapiro and Leon C. Woodson, The Boundary of Ordered Trees, Journal of Integer Sequences, Vol. 18 (2015), Article 15.5.8.
• Filippo Disanto, Some Statistics on the Hypercubes of Catalan Permutations, Journal of Integer Sequences, Vol. 18 (2015), Article 15.2.2.
• Filippo Disanto and Thomas Wiehe, Some instances of a sub-permutation problem on pattern avoiding permutations, arXiv preprint arXiv:1210.6908 [math.CO], 2012-2014.
• Filippo Disanto and Thomas Wiehe, On the sub-permutations of pattern avoiding permutations, Discrete Math., 337 (2014), 127-141.
• Manuel Flores, Yuta Kimura and Baptiste Rognerud, Combinatorics of quasi-hereditary structures, arXiv:2004.04726 [math.RT], 2020.
• Sela Fried and Toufik Mansour, Counting r X s rectangles in (Catalan) words, arXiv:2405.06962 [math.CO], 2024. See p. 11.
• Alice L.L. Gao, Sergey Kitaev, and Philip B. Zhang, On pattern avoiding indecomposable permutations, arXiv:1605.05490 [math.CO], 2016.
• N. S. S. Gu, N. Y. Li and T. Mansour, 2-Binary trees: bijections and related issues, Discr. Math., 308 (2008), 1209-1221.
• R. K. Guy, Catwalks, Sandsteps and Pascal Pyramids, J. Integer Seqs., Vol. 3 (2000), #00.1.6
• Guo-Niu Han, Enumeration of Standard Puzzles [broken link]
• Guo-Niu Han, Enumeration of Standard Puzzles [Cached copy]
• 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.
• Sergey Kitaev, Generalized pattern avoidance with additional restrictions, Sem. Lothar. Combinat. B48e (2003). See also arXiv:0205215 [math.CO], 2002.
• Sergey Kitaev and Toufik Mansour, Simultaneous avoidance of generalized patterns, arXiv:math/0205182 [math.CO], 2002.
• C. Krishnamachary and M. Bheemasena Rao, Determinants whose elements are Eulerian, prepared Bernoullian and other numbers, J. Indian Math. Soc., 14 (1922), 55-62, 122-138 and 143-146. [Annotated scanned copy]
• D. Nečas, Elementary models of low-pressure plasma polymerisation into nanofibrous mats, Phys. Scr. (2025) Vol. 100, 055601. See p. 10.
• Ran Pan and Jeffrey B. Remmel, Paired patterns in lattice paths, arXiv:1601.07988 [math.CO], 2016.
• A. Papoulis, A new method of inversion of the Laplace transform, Quart. Appl. Math 14 (1957), 405-414. [Annotated scan of selected pages]
• A. Papoulis, A new method of inversion of the Laplace transform, Quart. Applied Math. 14 (1956), 405ff.
• J.-B. Priez and A. Virmaux, Non-commutative Frobenius characteristic of generalized parking functions: Application to enumeration, arXiv:1411.4161 [math.CO], 2014-2015.
• Jocelyn Quaintance and Harris Kwong, A combinatorial interpretation of the Catalan and Bell number difference tables, Integers, 13 (2013), #A29.
• J. Riordan, Letter to N. J. A. Sloane, Nov 10 1970
• J. Riordan, The distribution of crossings of chords joining pairs of 2n points on a circle, Math. Comp., 29 (1975), 215-222.
• Zoran Sunic, Self describing sequences and the Catalan family tree, Elect. J. Combin., 10 (No. 1, 2003).
• Murray Tannock, Equivalence classes of mesh patterns with a dominating pattern, MSc Thesis, Reykjavik Univ., May 2016.
• S. J. Tedford, Combinatorial interpretations of convolutions of the Catalan numbers, Integers 11 (2011) #A3.
• Bridget E. Tenner, Interval structures in the Bruhat and weak orders, arXiv:2001.05011 [math.CO], 2020.
• Qi Wang, Tau-tilting finite simply connected algebras, arXiv:1910.01937 [math.RT], 2019.

Crossrefs

First differences of Catalan numbers A000108.

T(n, n+3) for n=0, 1, 2, ..., array T as in A047072.

Cf. A099364.

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

Cf. A002057, A000344, A003517, A000588, A003518, A003519, A001392, A001622.

Cf. A154558, A014137.

Column k=1 of A067323.

Programs

GAP

Concatenation([0],List([1..30],n->3*Factorial(2*n)/(Factorial(n+2)*Factorial(n-1)))); # Muniru A Asiru, Aug 09 2018

Magma

[0] cat [3*Factorial(2*n)/(Factorial(n+2)*Factorial(n-1)): n in [1..30]]; // Vincenzo Librandi, Feb 15 2016

Maple

A000245 := n -> 3*binomial(2*n, n-1)/(n+2);
seq(A000245(n), n=0..27);

Mathematica

Table[3(2n)!/((n+2)!(n-1)!),{n,0,30}] (* or *) Table[3*Binomial[2n,n-1]/(n+2),{n,0,30}] (* or *) Differences[CatalanNumber[Range[0,31]]] (* Harvey P. Dale, Jul 13 2011 *)

PARI

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

Sage

[catalan_number(i+1) - catalan_number(i) for i in range(28)] # Zerinvary Lajos, May 17 2009

Sage

def A000245_list(n) :
    D = [0]*(n+1); D[1] = 1
    b = False; h = 1; R = []
    for i in range(2*n-1) :
        if b :
            for k in range(h,0,-1) : D[k] += D[k-1]
            h += 1; R.append(D[2])
        else :
            for k in range(1,h, 1) : D[k] += D[k+1]
        b = not b
    return R
A000245_list(29) # Peter Luschny, Jun 03 2012

Formula

a(n) = A000108(n+1) - A000108(n).

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). - Wolfdieter Lang

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

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

G.f. satisfies x^2*A(x)^2 + (3*x-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, 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, 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)-2*C(1,k)+C(0, k))*(1+(-1)^(n-k))/(n+k+2). - Paul Barry, Oct 13 2004

a(n) = Sum_{k=0..n} binomial(n,k)*2^(n-k)*(-1)^(k+1)*binomial(k, floor((k-1)/2)). - Paul Barry, Feb 16 2006

E.g.f.: exp(2*x)*(Bessel_I(1,2x) - Bessel_I(2,2*x)). - Paul Barry, Jun 04 2007

a(n) = (1/Pi)*Integral_{x=0..4} x^n*(x-1)*sqrt(x*(4-x))/(2*x). - Paul Barry, Feb 08 2008

D-finite with recurrence: For n > 1, a(n+1) = 2*(2n+1)*(n+1)*a(n)/((n+3)*n). - Sean A. Irvine, Dec 09 2009

Let A be the Toeplitz matrix of order n defined by: A[i,i-1]=-1, A[i,j] = Catalan(j-i), (i<=j), and A[i,j] = 0, otherwise. Then, for n >= 2, a(n-1) = (-1)^(n-2)*coeff(charpoly(A,x),x^2). - Milan Janjic, Jul 08 2010

a(n) = sum of top row terms of M^(n-1), M = an infinite square production matrix as follows:

2, 1, 0, 0, 0, 0, ...

1, 1, 1, 0, 0, 0, ...

1, 1, 1, 1, 0, 0, ...

1, 1, 1, 1, 1, 0, ...

1, 1, 1, 1, 1, 1, ...

...

- Gary W. Adamson, Jul 14 2011

E.g.f.: exp(2*x)*(BesselI(2,2*x)) = Q(0) - 1 where Q(k) = 1 - 2*x/(k + 1 - 3*((k+1)^2)/((k^2) + 8*k + 9 - (k+2)*((k+3)^2)*(2*k+3)/((k+3)*(2*k+3) - 3*(k+1)/Q(k+1)))); (continued fraction). - Sergei N. Gladkovskii, Dec 05 2011

a(n) = -binomial(2*n,n)/(n+1)*hypergeom([-1,n+1/2],[n+2],4). - Peter Luschny, Aug 15 2012

a(n) = Sum_{i=0..n-1} C(i)*C(n-i), where C(i) denotes the i-th Catalan number. - Dmitry Kruchinin, Mar 02 2013

a(n) = binomial(2*n-1, n) - binomial(2*n-1, n-3). - Johannes W. Meijer, Jul 31 2013

a(n) ~ 3*4^n/(n^(3/2)*sqrt(Pi)). - Vaclav Kotesovec, Feb 26 2016

a(n) = ((-1)^n/(n+1))*Sum_{i=0..n-1} (-1)^(i+1)*(n+1-i)*binomial(2*n+2,i), n>=0. - Taras Goy, Aug 09 2018

From Amiram Eldar, Jan 02 2022: (Start)

Sum_{n>=1} 1/a(n) = 14*Pi/(27*sqrt(3)) + 5/9.

Sum_{n>=1} (-1)^(n+1)/a(n) = 164*log(phi)/(75*sqrt(5)) + 7/25, where phi is the golden ratio (A001622). (End)

a(n) = 3*Sum_{k = 0..n-2} (-1)^k * binomial(2*n-k-1, n+1)*binomial(n+1, k)/(k + 1) for n >= 2. - Peter Bala, Sep 02 2024

a(n) = (3*n/(n+2))*A000108(n). - Taras Goy, Dec 20 2024

Extensions

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, Oct 31 2003