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We appear to get the same sequence by expanding 1 - (1 - exp(t)) + (1 - exp(t))*(1 - exp(2*t)) - (1 - exp(t))*(1 - exp(2*t))*(1 - exp(3*t)) + ... as a series in t. Compare with A079144. For other sequences with generating functions of a similar type see A000364, A000464, A002105 and A002439.

From Peter Bala, Mar 13 2017: (Start)

It appears that the g.f. has two other forms: either F(exp(-t)) where F(q) = Sum_{n >= 0} q^(n+1)*Product_{k = 1..n} 1 - q^(2*k) = q + q^2 + q^3 - q^7 - q^8 - q^10 - q^11 - ... is a g.f. for A003475 or 1/2*G(exp(t)) where G(q) = 1 + Sum_{n >= 0} (-1)^n*q^(n+1)*Product_{k = 1..n} 1 - q^k = 1 + q - q^2 + 2*q^3 - 2*q^4 + q^5 + q^7 - 2*q^8 + ... is a g.f. for A003406. See Zagier, Example 1. (End)

From Peter Bala, Dec 18 2021: (Start)

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 16 begins [1, 1, 5, 7, 1, 15, 1, 15, 1, 15, 1, 15, ...] with an apparent pre-period of length 4 and a period of length 2.

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. For example, the expansion of exp( Sum_{n >= 1} a(n)*x^n/n ) = 1 + x + 3*x^2 + 21*x^3 + 291*x^4 + 6861*x^5 + 246171*x^6 + 12458901*x^7 + 843915891*x^8 + 73640674461*x^9 + 8041227405771*x^10 + ... appears to have integer coefficients. (End)

From Vít Jelínek, Feb 12 2012: (Start)

a(n) has the following combinatorial interpretations:

(1) the number of upper-triangular matrices over {0,1} having at least one '1'-entry in each row and having n '1'-entries in total. E.g., for n=2, this corresponds to these two matrices (with zeros represented as dots):

1. .1

.1 .1

(2) the number of upper-triangular matrices over {0,1} that are symmetric with respect to the northeast diagonal, have at least one '1'-entry in each row and column, have no '1'-entry on the northeast diagonal, and have 2n '1'-entries in total. For n=2, those are the two matrices

11. 1...

..1 .1..

..1 ..1.

...1

(3) the number of upper-triangular matrices over {0,1} that are symmetric with respect to the northeast diagonal, have at least one '1'-entry in each row and column, have at least one '1'-entry on the northeast diagonal, and have n '1'-entries on or above the northeast diagonal. For n=2, this corresponds to

11 1..

.1 .1.

..1

(End)

This is an example of Peter Bala's identity (cf. A158690):

Sum_{n>=0} Product_{k=1..n} (q^k - 1) = Sum_{n>=0} q^(-n^2) * Product_{k = 1..n} (q^(2*k-1) - 1) at q = 1 + x. See cross-references for other examples.

Motivated by Peter Bala's identity described in A158690:

Sum_{n>=0} Product_{k=1..n} (q^k - 1) =

Sum_{n>=0} q^(-n^2) * Product_{k=1..n} (q^(2*k-1) - 1),

here q = (1+x)/(1+x^3). See cross-references for other examples.

At present Bala's identity is conjectural and needs formal proof.

Compare g.f. to Sum_{n>=0} Product_{k=1..n} ((1+x)^k - 1), which is the g.f. of A179525.

Compare g.f. to Sum_{n>=0} Product_{k=1..n} (1 - (1 - x)^(2*k-1)), which is the g.f. of A158691. - Peter Bala, Dec 04 2020

Motivated by Peter Bala's identity described in A158690:

Sum_{n>=0} Product_{k=1..n} (q^k - 1) =

Sum_{n>=0} q^(-n^2) * Product_{k=1..n} (q^(2*k-1) - 1),

here q = (1+x)/(1+x^2). See cross-references for other examples.

At present Bala's identity is conjectural and needs formal proof.

Motivated by Peter Bala's identity described in A158690:

Sum_{n>=0} Product_{k=1..n} (q^k - 1) =

Sum_{n>=0} q^(-n^2) * Product_{k=1..n} (q^(2*k-1) - 1),

here q = (1+x)/(1+x^2). See cross-references for other examples.

At present Bala's identity is conjectural and needs formal proof.

a(n) = number of upper triangular matrices with entries from {0,1,2} with no zero rows such that the sum of the entries is n, that is, row Fishburn matrices of size n with entries from {0,1,2}. Cf. A179525. - Peter Bala, Nov 05 2017

A row-Fishburn matrix of size n is defined to be an upper-triangular matrix with nonnegative integer entries which sum to n and such that each row contains a nonzero entry. See A158691.

Here we consider generalized row-Fishburn matrices where we allow the row_Fishburn matrices to have positive and negative nonzero entries. We define the size of a generalized row-Fishburn matrix to be the absolute sum of the matrix entries. This sequence gives the number of generalized row-Fishburn matrices of size n.

Alternatively, this sequence gives the number of 2-colored row-Fishburn matrices of size n, that is, ordinary row-Fishburn matrices of size n where each nonzero entry in the matrix can have one of two different colors.

More generally, if F(x) = Sum_{n >= 0} ( Product_{i = 1..n} (1 + x)^i - 1 ) is the o.g.f. for primitive row-Fishburn matrices A179525 (i.e., row-Fishburn matrices with entries restricted to the set {0,1}) and C(x) := c_1*x + c_2*x^2 + ..., where c_i is a sequence of nonnegative integers, then the composition F(C(x)) is the o.g.f. for colored row-Fishburn matrices where entry i in the matrix can have one of c_i different colors: c_i = 0 for some i means i does not appear as an entry in the Fishburn matrix. This result is an application of Lemma 2.2.22 of Goulden and Jackson.

A row-Fishburn matrix of size n is defined to be an upper-triangular matrix with nonnegative integer entries which sum to n and each row contains a nonzero entry. See A158691. Here we are considering row-Fishburn matrices where the nonzero entries are all odd.

The g.f. F(x) for primitive row_Fishburn matrices (i.e., row_Fishburn matrices with entries restricted to the set {0,1}), is F(x) = Sum_{n>=0} Product_{k=1..n} ( (1 + x)^k - 1 ). See A179525. Let C(x) = x/(1 - x^2) = x + x^3 + x^5 + x^7 + .... Then appplying Lemma 2.2.22 of Goulden and Jackson gives the g.f. for the present sequence as the composition F(C(x)).