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Equivalently, result of applying same transformation 4 times to g.f. for Catalan numbers A000108.

These numbers k have the smallest A010846(k) for a number with n distinct prime factors.

a(7) <= 1398483454696343742813089 = 1049 * 2819 * 3319 * 3433 * 3457 * 3463 * 3467.

a(8) <= 32829974457045619959776094471833047127947.

Terms in row n are sorted lexicographically.

Row n is created by finding n-tuples w with elements from 0..n-1, taking only those w whose sums are less than n.

For example, row n = 3 contains 3-tuples w that have elements from 0..2, i.e., {(0,0,0), (0,0,1), (0,0,2), (0,1,0), (0,1,1), (0,2,0), (1,0,0), (1,0,1), (1,1,0), (2,0,0)}.

Let s be the sum of w. Then we take all elements w of row n-1 and append n-s to w to obtain certain 3-tuples with elements from 0..n whose sum s = n.

Continuing the example, row 2 = {(0,0), (0,1), (0,2), (1,0), (1,1)}, which, adding n-s to the right end gives {(0,0,3), (0,1,2), (0,2,1), (1,0,2), (1,1,1)}.

Let p_i be the i-th smallest prime divisor of N = A384000(n) (where i is not necessarily the i-th prime). Then, the terms m in row N of A162306 are of the form m = Product_{i..n} p_i^T(n,j,n-k+1). For instance, for N = 6, we have row 6 of A162306 = {1, 2, 3, 4, 6}, which is {2^0*3^0, 2^1*3^0, 2^2*3^0, 2^0*3^1, 2^1*3^1} = {1, 2, 4, 3, 6}, sorted.

Essentially the same as A001700, which has more information.

Note that the unique rooted tree with no edges has no leaves, so a(0)=1 is by convention. - Michael Somos, Jul 30 2011

Number of ordered partitions of n into n parts, allowing zeros (cf. A097070) is binomial(2*n-1,n) = a(n) = essentially A001700. - Vladeta Jovovic, Sep 15 2004

Hankel transform is A000027; example: Det([1,1,3,10;1,3,10,35;3,10,35,126; 10,35,126,462]) = 4. - Philippe Deléham, Apr 13 2007

a(n) is the number of functions f:[n]->[n] such that for all x,y in [n] if xA045992(n). - Geoffrey Critzer, Apr 02 2009

Hankel transform of the aeration of this sequence is A000027 doubled: 1,1,2,2,3,3,... - Paul Barry, Sep 26 2009

The Fi1 and Fi2 triangle sums of A039599 are given by the terms of this sequence. For the definitions of these triangle sums see A180662. - Johannes W. Meijer, Apr 20 2011

Alternating row sums of Riordan triangle A094527. See the Philippe Deléham formula. - Wolfdieter Lang, Nov 22 2012

(-2)*a(n) is the Z-sequence for the Riordan triangle A110162. For the notion of Z- and A-sequences for Riordan arrays see the W. Lang link under A006232 with details and references. - Wolfdieter Lang, Nov 22 2012

From Gus Wiseman, Jun 27 2021: (Start)

Also the number of integer compositions of 2n with alternating (or reverse-alternating) sum 0 (ranked by A344619). This is equivalent to Ran Pan's comment at A001700. For example, the a(0) = 1 through a(3) = 10 compositions are:

() (11) (22) (33)

(121) (132)

(1111) (231)

(1122)

(1221)

(2112)

(2211)

(11121)

(12111)

(111111)

For n > 0, a(n) is also the number of integer compositions of 2n with alternating sum 2.

(End)

Number of terms in the expansion of (x_1+x_2+...+x_n)^n. - César Eliud Lozada, Jan 08 2022

This sequence, with first term a(0) deleted, appears to be determined by the conditions that the diagonal and first superdiagonal of U are {1,1,1,1,...} and {2,3,4,5,...,n+1,...} respectively, where A=LU is the LU factorization of the Hankel matrix A given by [{a(1),a(2),...}, {a(2),a(3),...}, ..., {a(n),a(n+1),...}, ...]. - John W. Layman, Jul 21 2000

Also the number of base 3 n-digit numbers (not starting with 0) with digit sum n. For the analogous sequence in base 10 see A071976, see example. - John W. Layman, Jun 22 2002

Also number of paths in an n X n grid from (0,0) to the line x=n-1, using only steps U=(1,1), H=(1,0) and D=(1,-1) (i.e., left factors of length n-1 of Motzkin paths, palindromic Motzkin paths of length 2n-2 or 2n-1). Example: a(3)=5, namely, HH, UD, HU, UH and UU. Also number of ordered trees with n edges and having nonroot nodes of outdegree at most 2. - Emeric Deutsch, Aug 01 2002

Number of symmetric Dyck paths of semilength 2n-1 with no peaks at even level. Example: a(3)=5 because we have UDUDUDUDUD, UDUUUDDDUD, UUUUUDDDDD, UUUDUDUDDD and UUUDDUUDDD, where U=(1,1) and D=(1,-1). Also number of symmetric Dyck paths of semilength 2n with no peaks at even level. Example: a(3)=5 because we have UDUDUDUDUDUD, UDUUUDUDDDUD, UUUDUDUDUDDD, UUUUUDUDDDDD and UUUDDDUUUDDD. - Emeric Deutsch, Nov 21 2003

a(n) is the sum of the (n-1)-st central trinomial coefficient and its predecessor. Example: a(4) = 6 + 7 and (1 + x + x^2)^3 = ... + 6*x^2 + 7*x^3 + ... . - David Callan, Feb 07 2004

a(n) is the number of UDU-free paths of n upsteps (U) and n downsteps (D) that start U (n>=1). Example: a(2)=2 counts UUDD, UDDU. - David Callan, Aug 18 2004

a(n) is also the number of Grand-Dyck paths of semilength n starting with an up-step and avoiding the pattern DUD. - David Bevan, Nov 19 2019

Hankel transform of a(n+1) = [1,2,5,13,35,96,...] gives A000012 = [1,1,1,1,1,1,...]. - Philippe Deléham, Oct 24 2007

Equals row sums of triangle A136787 starting (1, 2, 5, 13, 35, ...). - Gary W. Adamson, Jan 21 2008

a(n) is the number of permutations on [n] that avoid the patterns 1-23-4 and 1-3-2, where the omission of a dash in a pattern means the permutation entries must be adjacent. Example: a(4) = 13 counts all 14 (Catalan number) (1-3-2)-avoiding permutations on [4] except 1234. - David Callan, Jul 22 2008

a(n) is also the number of involutions of length 2n-2 which are invariant under the reverse-complement map and have no decreasing subsequences of length 4. - Eric S. Egge, Oct 21 2008

Hankel transform is A010892. - Paul Barry, Jan 19 2009

a(n) is the number of Dyck words of semilength n with no DUUU. For example, a(4) = 14-1 = 13 because there is only one Dyck 4-word containing DUUU, namely UDUUUDDD. - Eric Rowland, Apr 21 2009

Inverse binomial transform of A024718. - Philippe Deléham, Dec 13 2009

Let w(i, j, n) denote walks in N^2 which satisfy the multivariate recurrence

w(i, j, n) = w(i - 1, j, n - 1) + w(i, j - 1, n - 1) + w(i + 1, j - 1,n - 1) with boundary conditions w(0,0,0) = 1 and w(i,j,n) = 0 if i or j or n is < 0. Let alpha(n) the number of such walks of length n, alpha(n) = Sum_{i = 0..n, j=0..n} w(i, j, n). Then a(n+1) = alpha(n). - Peter Luschny, May 21 2011

Number of length-n strings [d(0),d(1),d(2),...,d(n-1)] where 0 <= d(k) <= k and abs(d(k) - d(k-1)) <= 1 (smooth factorial numbers, see example). - Joerg Arndt, Nov 10 2012

a(n) is the number of n-multisets of {1,...,n} containing no pair of consecutive integers (e.g., 111, 113, 133, 222, 333 for n=3). - David Bevan, Jun 10 2013

a(n) is also the number of n-multisets of [n] in which no integer except n occurs exactly once (e.g., 111, 113, 222, 223, 333 for n=3). - David Bevan, Nov 19 2019

Number of minimax elements in the affine Weyl group of the Lie algebra so(2n+1) or the Lie algebra sp(2n). See Panyushev 2005. Cf. A245455. - Peter Bala, Jul 22 2014

The shifted, signed array belongs to an interpolated family of arrays associated to the Catalan A000108 (t=1), and Riordan, or Motzkin sums A005043 (t=0), with the interpolating (here t=-2) o.g.f. G(x,t) = (1-sqrt(1-4x/(1+(1-t)x)))/2 and inverse o.g.f. Ginv(x,t) = x(1-x)/(1+(t-1)x(1-x)) (A057682). See A091867 for more info on this family. - Tom Copeland, Nov 09 2014

Alternatively, this sequence corresponds to the number of positive walks with n steps {-1,0,1} starting at the origin, ending at any altitude, and staying strictly above the x-axis. - David Nguyen, Dec 01 2016

Let N be a squarefree number with n prime factors: p_1 < p_2 < ... < p_n. Let D be its set of divisors, E the subset of D X D made of the (d_1, d_2) for which, provided that we know which p_i are in d_1, which p_i are in d_2, d_1 <= d_2 is provable without needing to know the numerical values of the p_i. It appears that a(n+1) is the number of (d_1, d_2) in E such that d_1 and d_2 are coprime. - Luc Rousseau, Aug 21 2017

Number of ordered rooted trees with n non-root nodes and all non-root nodes having outdegrees 1 or 2. - Andrew Howroyd, Dec 04 2017

a(n) is the number of compositions (ordered partitions) of n where there are A001006(k-1) sorts of part k (see formula by Andrew Howroyd, Dec 04 2017). - Joerg Arndt, Jan 26 2024

The expression a(n) = B^n*Sum_{ k=0..n } binomial(2*k,k)/B^k gives A006134 for B=1, A082590 (B=2), A132310 (B=3), A002457 (B=4), A144635 (B=5). - N. J. A. Sloane, Jan 21 2009

T(n+1,1) from table A045912 of characteristic polynomial of negative Pascal matrix. - Michael Somos, Jul 24 2002

p divides a((p-3)/2) for p=11, 13, 23, 37, 47, 59, 61, 71, 73, 83, 97, 107, 109, 131, 157, 167, ...: A097933. Also primes congruent to {1, 2, 3, 11} mod 12 or primes p such that 3 is a square mod p (excluding 2 and 3) A038874. - Alexander Adamchuk, Jul 05 2006

Partial sums of the even central binomial coefficients. For p prime >=5, a(p-1) = 1 or -1 (mod p) according as p = 1 or -1 (mod 3) (see Pan and Sun link). - David Callan, Nov 29 2007

First column of triangle A187887. - Michel Marcus, Jun 23 2013

From Gus Wiseman, Apr 20 2023: (Start)

Also the number of nonempty subsets of {1,...,2n+1} with median n+1, where the median of a multiset is either the middle part (for odd length), or the average of the two middle parts (for even length). The odd/even-length cases are A000984 and A006134(n-1). For example, the a(0) = 1 through a(2) = 9 subsets are:

{1} {2} {3}

{1,3} {1,5}

{1,2,3} {2,4}

{1,3,4}

{1,3,5}

{2,3,4}

{2,3,5}

{1,2,4,5}

{1,2,3,4,5}

Alternatively, a(n-1) is the number of nonempty subsets of {1,...,2n-1} with median n.

(End)

a(n) is the sum of pyramid weights of all Dyck paths of length 2n (for pyramid weight see Denise and Simion). Equivalently, a(n) is the sum of the total lengths of end branches of an ordered tree, summation being over all ordered trees with n edges. For example, the five ordered trees with 3 edges have total lengths of endbranches 3,2,3,3 and 3. - Emeric Deutsch, May 30 2003

a(n) is the number of Motzkin paths of length 2n with exactly one level segment. (A level segment is a maximal sequence of contiguous flatsteps.) Example: for n=2, the paths counted are FFFF, FFUD, UDFF, UFFD. The formula for a(n) below counts these paths by length of the level segment. - David Callan, Jul 15 2004

The inverse Catalan transform yields A024495, shifted once left. - R. J. Mathar, Jul 07 2009

From Paul Barry, Mar 29 2010: (Start)

Hankel transform is A138341.

The aerated sequence 0, 0, 1, 0, 4, 0, 14, 0, 49, ... has e.g.f. int(cosh(x-t)*Bessel_I(1,2t), t = 0..x). (End)

a(n) is the number of terms of A031443 not exceeding 4^n. - Vladimir Shevelev, Oct 01 2010

Also the number of nonempty subsets of {1..2n} with median n, bisection of A361801. The version containing n is A001700 (bisected). Replacing 2n with 2n+1 and n with n+1 gives A006134. For mean instead of median we have A212352. - Gus Wiseman, Apr 16 2023

The 4-term sequence 2,4,8,20 is the answer to the "Solitaire Army" problem, or checker-jumping puzzle. It is too short to have its own entry. See Conway et a;., Winning Ways, Vol. 2, pp. 715-717. - N. J. A. Sloane, Mar 01 2018

Number of subsets of {-n..n} with sum 0. Also the number of subsets of {0..2n} that are empty or have mean n. For median instead of mean we have twice A024718. - Gus Wiseman, Apr 23 2023

From Gus Wiseman, Apr 19 2023: (Start)

Appears to be the number of nonempty subsets of {1,...,n} with median k, where the median of a multiset is either the middle part (for odd length), or the average of the two middle parts (for even length). For example, row n = 5 counts the following subsets:

{1} {2} {3} {4} {5}

{1,3} {1,5} {3,5}

{1,2,3} {2,4} {1,4,5}

{1,2,4} {1,3,4} {2,4,5}

{1,2,5} {1,3,5} {3,4,5}

{2,3,4}

{2,3,5}

{1,2,4,5}

{1,2,3,4,5}

Including half-steps gives A231147.

For mean instead of median we have A327481.

(End)