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A000245 a(n) = 3*(2*n)!/((n+2)!*(n-1)!).
(Formerly M2809 N1130)

%I M2809 N1130

%S 0,1,3,9,28,90,297,1001,3432,11934,41990,149226,534888,1931540,

%T 7020405,25662825,94287120,347993910,1289624490,4796857230,

%U 17902146600,67016296620,251577050010,946844533674,3572042254128,13505406670700,51166197843852,194214400834356

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

%C 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

%C 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

%C 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

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

%C 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

%C a(n)=number of Dyck (n+1)-paths that start with UU. For example, a(2)=3 counts UUUDDD, UUDUDD, UUDDUD. - _David Callan_, Dec 08 2004

%C a(n)=number of Dyck (n+2)-paths that start with UUDU. For example, a(2)=3 counts UUDUDDUD, UUDUDUDD, UUDUUDDD. - _David Scambler_, Feb 14 2011

%C 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

%C Starting with offset 1 = row sums of triangle A154558. - _Gary W. Adamson_, Jan 11 2009

%C 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

%C 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

%C 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

%C a(n+2) = A214292(2*n+3,n). - _Reinhard Zumkeller_, Jul 12 2012

%C 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

%C 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

%C 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

%C 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

%D P. de la Harpe, Topics in Geometric Group Theory, Univ. Chicago Press, 2000, p. 11, coefficients of P_3(z).

%D Kim, Ki Hang; Rogers, Douglas G.; Roush, Fred W. 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)

%D 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.

%D N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

%H T. D. Noe, <a href="/A000245/b000245.txt">Table of n, a(n) for n = 0..100</a>

%H Marco Abrate, Stefano Barbero, Umberto Cerruti, Nadir Murru, <a href="http://dx.doi.org/10.1016/j.disc.2014.06.026">Colored compositions, Invert operator and elegant compositions with the "black tie"</a>, Discrete Math. 335 (2014), 1-7. MR3248794

%H J.-L. Baril, <a href="http://www.combinatorics.org/ojs/index.php/eljc/article/view/v18i1p178">Classical sequences revisited with permutations avoiding dotted pattern</a>, Electronic Journal of Combinatorics, 18 (2011), #P178.

%H J.-L. Baril, S. Kirgizov, <a href="http://jl.baril.u-bourgogne.fr/Stirling.pdf">The pure descent statistic on permutations</a>, Preprint, 2016.

%H Paul Barry, <a href="http://www.cs.uwaterloo.ca/journals/JIS/VOL8/Barry/barry84.html">A Catalan Transform and Related Transformations on Integer Sequences</a>, Journal of Integer Sequences, Vol. 8 (2005), Article 05.4.5.

%H Paul Barry, <a href="https://arxiv.org/abs/1807.05794">Riordan Pseudo-Involutions, Continued Fractions and Somos 4 Sequences</a>, arXiv:1807.05794 [math.CO], 2018.

%H D. Callan, <a href="http://arXiv.org/abs/math.CO/0211380">A recursive bijective approach to counting permutations...</a>, arXiv:math/0211380 [math.CO], Nov 25 2002.

%H S. Connolly, Z. Gabor and A. Godbole, <a href="http://arxiv.org/abs/1401.2691"> The location of the first ascent in a 123-avoiding permutation</a>, arXiv:1401.2691 [math.CO], 2014.

%H S. J. Cyvin and I. Gutman, <a href="https://doi.org/10.1007/978-3-662-00892-8">Kekulé structures in benzenoid hydrocarbons</a>, Lecture Notes in Chemistry, No. 46, Springer, New York, 1988 (see p. 196).

%H Dennis E. Davenport, Lara K. Pudwell, Louis W. Shapiro, Leon C. Woodson, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL18/Davenport/dav3.html">The Boundary of Ordered Trees</a>, Journal of Integer Sequences, Vol. 18 (2015), Article 15.5.8.

%H Filippo Disanto, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL18/Disanto/disanto5.html">Some Statistics on the Hypercubes of Catalan Permutations</a>, Journal of Integer Sequences, Vol. 18 (2015), Article 15.2.2.

%H Filippo Disanto and Thomas Wiehe, <a href="http://arxiv.org/abs/1210.6908">Some instances of a sub-permutation problem on pattern avoiding permutations</a>, arXiv preprint arXiv:1210.6908 [math.CO], 2012-2014.

%H F. Disanto and T. Wiehe, <a href="http://dx.doi.org/10.1016/j.disc.2014.06.028">On the sub-permutations of pattern avoiding permutations</a>, Discrete Math., 337 (2014), 127-141.

%H A. L. L. Gao, S. Kitaev, P. B. Zhang. <a href="https://arxiv.org/abs/1605.05490">On pattern avoiding indecomposable permutations</a>, arXiv:1605.05490 [math.CO], 2016.

%H N. S. S. Gu, N. Y. Li and T. Mansour, <a href="http://dx.doi.org/10.1016/j.disc.2007.04.007">2-Binary trees: bijections and related issues</a>, Discr. Math., 308 (2008), 1209-1221.

%H R. K. Guy, <a href="http://www.cs.uwaterloo.ca/journals/JIS/VOL3/GUY/catwalks.html">Catwalks, Sandsteps and Pascal Pyramids</a>, J. Integer Seqs., Vol. 3 (2000), #00.1.6

%H Guo-Niu Han, <a href="http://www-irma.u-strasbg.fr/~guoniu/papers/p77puzzle.pdf">Enumeration of Standard Puzzles</a> [broken link]

%H Guo-Niu Han, <a href="/A196265/a196265.pdf">Enumeration of Standard Puzzles</a> [Cached copy]

%H V. E. Hoggatt, Jr. and M. Bicknell, <a href="http://www.fq.math.ca/Scanned/14-5/hoggatt1.pdf">Catalan and related sequences arising from inverses of Pascal's triangle matrices</a>, Fib. Quart., 14 (1976), 395-405.

%H S. Kitaev, <a href="http://www.mat.univie.ac.at/users/slc/public_html/wpapers/s48kitaev.html">Generalized pattern avoidance with additional restrictions</a>, Sem. Lothar. Combinat. B48e (2003).

%H S. Kitaev and T. Mansour, <a href="http://arXiv.org/abs/math.CO/0205182">Simultaneous avoidance of generalized patterns</a>, arXiv:math/0205182 [math.CO], 2002.

%H C. Krishnamachary and M. Bheemasena Rao, <a href="/A000108/a000108_10.pdf">Determinants whose elements are Eulerian, prepared Bernoullian and other numbers</a>, J. Indian Math. Soc., 14 (1922), 55-62, 122-138 and 143-146. [Annotated scanned copy]

%H Ran Pan, Jeffrey B. Remmel, <a href="http://arxiv.org/abs/1601.07988">Paired patterns in lattice paths</a>, arXiv:1601.07988 [math.CO], 2016.

%H A. Papoulis, <a href="/A000108/a000108_8.pdf">A new method of inversion of the Laplace transform</a>, Quart. Appl. Math 14 (1957), 405-414. [Annotated scan of selected pages]

%H A. Papoulis, <a href="http://www.jstor.org/stable/43636019">A new method of inversion of the Laplace transform</a>, Quart. Applied Math. 14 (1956), 405ff.

%H J.-B. Priez, A. Virmaux, <a href="http://arxiv.org/abs/1411.4161">Non-commutative Frobenius characteristic of generalized parking functions: Application to enumeration</a>, arXiv:1411.4161 [math.CO], 2014-2015.

%H Jocelyn Quaintance and Harris Kwong, <a href="http://www.emis.de/journals/INTEGERS/papers/n29/n29.Abstract.html">A combinatorial interpretation of the Catalan and Bell number difference tables</a>, Integers, 13 (2013), #A29.

%H J. Riordan, <a href="/A000262/a000262_1.pdf">Letter to N. J. A. Sloane, Nov 10 1970</a>

%H J. Riordan, <a href="http://www.jstor.org/stable/2005477">The distribution of crossings of chords joining pairs of 2n points on a circle</a>, Math. Comp., 29 (1975), 215-222.

%H Zoran Sunic, <a href="http://www.combinatorics.org/ojs/index.php/eljc/article/view/v10i1n5">Self describing sequences and the Catalan family tree</a>, Elect. J. Combin., 10 (No. 1, 2003).

%H Murray Tannock, <a href="https://skemman.is/bitstream/1946/25589/1/msc-tannock-2016.pdf">Equivalence classes of mesh patterns with a dominating pattern</a>, MSc Thesis, Reykjavik Univ., May 2016.

%H S. J. Tedford, <a href="http://www.emis.de/journals/INTEGERS/papers/l3/l3.Abstract.html">Combinatorial interpretations of convolutions of the Catalan numbers</a>, Integers 11 (2011) #A3.

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

%F 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_

%F 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

%F 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.

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

%F Series reversion of g.f. A(x) is -A(-x). - _Michael Somos_, Jan 21 2004

%F 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

%F An inverse Chebyshev transform of x^2. - _Paul Barry_, Oct 13 2004

%F 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

%F 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

%F 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

%F E.g.f.: exp(2*x)*(Bessel_I(1,2x) - Bessel_I(2,2*x)). - _Paul Barry_, Jun 04 2007

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

%F For n > 1, a(n+1) = 2*(2n+1)*(n+1)*a(n)/((n+3)*n). - _Sean A. Irvine_, Dec 09 2009

%F 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

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

%F 2, 1, 0, 0, 0, 0, ...

%F 1, 1, 1, 0, 0, 0, ...

%F 1, 1, 1, 1, 0, 0, ...

%F 1, 1, 1, 1, 1, 0, ...

%F 1, 1, 1, 1, 1, 1, ...

%F ...

%F - _Gary W. Adamson_, Jul 14 2011

%F 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

%F a(n) = -binomial(2*n,n)/(n+1)*hypergeom([-1,n+1/2],[n+2],4). - _Peter Luschny_, Aug 15 2012

%F 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

%F a(n) = binomial(2*n-1, n) - binomial(2*n-1, n-3). - _Johannes W. Meijer_, Jul 31 2013

%F a(n) ~ 3*4^n/(n^(3/2)*sqrt(Pi)). - _Vaclav Kotesovec_, Feb 26 2016

%F 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

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

%p seq(A000245(n), n=0..27);

%t 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 *)

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

%o (Sage) [catalan_number(i+1)-catalan_number(i) for i in xrange(0,28)] # _Zerinvary Lajos_, May 17 2009

%o (Sage)

%o def A000245_list(n) :

%o D = [0]*(n+1); D[1] = 1

%o b = False; h = 1; R = []

%o for i in range(2*n-1) :

%o if b :

%o for k in range(h,0,-1) : D[k] += D[k-1]

%o h += 1; R.append(D[2])

%o else :

%o for k in range(1,h, 1) : D[k] += D[k+1]

%o b = not b

%o return R

%o A000245_list(29) # _Peter Luschny_, Jun 03 2012

%o (MAGMA) [0] cat [3*Factorial(2*n)/(Factorial(n+2)*Factorial(n-1)): n in [1..30]]; // _Vincenzo Librandi_, Feb 15 2016

%o (GAP) Concatenation([0],List([1..30],n->3*Factorial(2*n)/(Factorial(n+2)*Factorial(n-1)))); # _Muniru A Asiru_, Aug 09 2018

%Y First differences of Catalan numbers A000108.

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

%Y Also a diagonal of A059365 and of A009766.

%Y Cf. A099364.

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

%Y Cf. A002057, A000344, A003517, A000588, A003518, A003519, A001392.

%Y Cf. A154558, A014137.

%Y Column k=1 of A067323.

%K nonn,easy,nice,changed

%O 0,3

%A _N. J. A. Sloane_

%E 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

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Last modified January 19 20:11 EST 2019. Contains 319309 sequences. (Running on oeis4.)