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Claesson and Linusson calls these the Fishburn numbers, after Peter Fishburn. [Peter Clingerman Fishburn (1936-2021) was an American mathematician and a pioneer in the field of decision-making processes. - Amiram Eldar, Jun 20 2021]

Also, number of unlabeled (2+2)-free posets.

Also, number of upper triangular matrices with nonnegative integer entries and without zero rows or columns such that sum of all entries is equal to n. - Vladeta Jovovic, Mar 10 2008

Row sums of A193387.

Also number of ascent sequences of length n. - N. J. A. Sloane, Dec 10 2011

An ascent sequence is a sequence [d(1), d(2), ..., d(n)] where d(1)=0, d(k)>=0, and d(k) <= 1 + asc([d(1), d(2), ..., d(k-1)]) where asc(.) counts the ascents of its argument. - Joerg Arndt, Oct 17 2012

Replacing the function asc(.) above by a function chg(.) that counts changes in the prefix gives A000522 (arrangements). - Joerg Arndt, May 10 2013

Number of restricted growth strings (RGS) [d(0), d(1), d(2), ..., d(n-1)] such that d(0)=0, 0 <= d(k) <= k, and d(j) != d(k)+1 for j < k. - Joerg Arndt, May 10 2013

Number of permutations p(1),p(2),...,p(n) having no subsequence p(i),p(j),p(k) such that i+1 = j < k and p(k)+1 = p(i) < p(j); this corresponds to the avoidance of bivincular pattern (231,{1},{1}) in the terminology of Parviainen. Also, number of permutations p(1),...,p(n) with no subsequence p(i),p(j),p(k) with i+1 = j < k, p(i)+1 = p(k) < p(j). This is the bivincular pattern (132,{1},{1}). Proved by Bousquet-Mélou et al. and by Parviainen, respectively. - Vít Jelínek, Sep 05 2014

In general, column k is asymptotic to Pi^(2*k-5/2) / (k! * 6^(k-2) * sqrt(3) * exp(Pi^2/12)) * (6/Pi^2)^n * n! * sqrt(n). - Vaclav Kotesovec, Aug 27 2014

From Peter Bala, Dec 26 2021: (Start)

We make the following conjectures:

1) Taking the sequence modulo an integer k gives an eventually periodic sequence with period dividing phi(k). For example, the sequence taken modulo 8 begins [1, 1, 3, 3, 7, 3, 7, 3, 7, ...] and appears to have a pre-period of length 3 and a period of length 2 = (1/2)*phi(8).

2) Let i >= 0 and define a_i(n) = a(n+i). Then for each i the Gauss congruences a_i(n*p^k) == a_i(n*p^(k-1)) ( mod p^k ) hold for all prime p and positive integers n and k. If true, then for each i the expansion of exp( Sum_{n >= 1} a_i(n)*x^n/n ) has integer coefficients. (End)

There is a connection with the alternating series 1 - (1-x) + (1-x)(1-x^2) - (1-x)(1-x^2)(1-x^3) + .... Namely, if we replace x with 1/(1-x) in the partial sum 1 - (1-x) + (1-x)(1-x^2) - (1-x)(1-x^2)(1-x^3) + ... + (-1)^n(1-x)(1-x^2)...(1-x^n) and then expand about x = 0 we get a series whose first n+1 coefficients agree with the first n+1 terms of the present sequence.

From Vít Jelínek, Sep 04 2014: (Start)

The above remark, first conjectured by Bala, is a consequence of the identities satisfied by the generating function for a(n). More precisely, the generating function F(x)=Sum_{n>=0} a(n)x^n can be expressed by any of these three formulas:

F(x) = Sum_{n>=0} Product_{i=1..n} (1-(1-x)^(2*i-1))

F(x) = Sum_{n>=0} Product_{i=1..n} (1/(1-x)^i - 1)

F(x) = Sum_{n>=0} (1-x)^(n+1) Product_{i=1..n} (1-(1-x)^(2*i))

The first two formulas were conjectured to be equal by Bala. This was confirmed by Andrews and Jelínek, who also derived the third formula.

Bringmann, Li and Rhoades have independently derived the three formulas above, and additionaly, they proved that

F(x) = (1/2)*Sum_{n>=0} Product_{i=1..n} (1/(1+(1-x)^i)).

(End)

From Vít Jelínek, Feb 12 2012: (Start)

a(n) has the following combinatorial interpretations:

(1) the number of upper-triangular nonnegative integer matrices with at least one positive entry in each row, whose entries sum to n. E.g., for n=2 this corresponds to these three matrices (with zeros represented as dots):

(2), (1.) and (.1)

(.1) (.1)

(2) the number of upper-triangular nonnegative matrices that are symmetric along the northeast diagonal, have no positive entry on the northeast diagonal, have at least one positive entry in every row and column, and their entries sum to 2n. These are the three such matrices with n=2:

(2.), (11.) and (1...)

(.2) (..1) (.1..)

(..1) (..1.)

(...1)

(3) the number of upper-triangular nonnegative matrices that are symmetric along the northeast diagonal, have at least one positive entry on the northeast diagonal, have at least one positive entry in every row and column, and their entries on or above the northeast diagonal sum to n. These are the three such matrices with n=2:

(2), (11) and (1..)

(.1) (.1.)

(..1)

(End)

A Fishburn matrix is defined to be an upper-triangular matrix with nonnegative integer entries such that each row and column contains a nonzero entry. See A005321 for primitive Fishburn matrices of dimension n, that is, Fishburn matrices of dimension n with entries in the set {0,1}.

The present sequence has an alternative description as the number of primitive Fishburn matrices of dimension n where the 1's may be colored either black or white.

A Fishburn matrix is defined to be an upper-triangular matrix with nonnegative integer entries such that each row and column contains a nonzero entry. See A005321 for primitive Fishburn matrices of dimension n, that is, Fishburn matrices of dimension n with entries in the set {0,1}.

The present sequence has an alternative description as the number of primitive Fishburn matrices of dimension n where each entry equal to 1 can have one of three different colors. Cf. A289314.

First column (also row sums) of triangle in A099959.

Number of ascent sequences of length n without level steps and with alternating ascents and descents. a(6) = 8: 010101, 010102, 010103, 010201, 010202, 010203, 010212, 010213. - Alois P. Heinz, Oct 27 2017