A179525 -id:A179525 - cp's OEIS frontend

A158691 The number of upper-triangular matrices with at least one nonzero entry in each row and whose entries sum to n.

1, 1, 3, 12, 61, 380, 2815, 24213, 237348, 2612681, 31915787, 428481472, 6271362282, 99388642292, 1695614865711, 30984649882928, 603790447393402, 12498732438500663, 273902239550757626, 6334968666307580051, 154211723833861061644, 3941258052200287007636

Offset: 0

Peter Bala, Mar 24 2009

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)

			G.f.: A(x) = 1 + x + 3*x^2 + 12*x^3 + 61*x^4 + 380*x^5 + 2815*x^6 +...

Seiichi Manyama, Table of n, a(n) for n = 0..200
George E. Andrews, Vít Jelínek, On q-Series Identities Related to Interval Orders, arXiv:1309.6669 [math.CO], 2013.
George E. Andrews, Vít Jelínek, On q-Series Identities Related to Interval Orders, European Journal of Combinatorics, Volume 39, July 2014, 178-187.
Kathrin Bringmann, Yingkun Li, Robert C. Rhoades, Asymptotics for the number of row-Fishburn matrices, European Journal of Combinatorics, Volume 41, October 2014, Pages 183-196; preprint.
Hsien-Kuei Hwang, Emma Yu Jin, Asymptotics and statistics on Fishburn matrices and their generalizations, arXiv:1911.06690 [math.CO], 2019.
Florian Ingels, Romain Azaïs, Enumeration of Unordered Forests, arXiv:2003.08144 [cs.DM], 2020.
Vít Jelínek, Counting self-dual interval orders, arXiv:1106.2261 [math.CO], 2011.
Vít Jelínek, Counting general and self-dual interval orders, Journal of Combinatorial Theory, Series A, Volume 119, Issue 3, April 2012, pp. 599-614.
Sherry H. F. Yan and Yuexiao Xu, Self-dual interval orders and row-Fishburn matrices, arXiv:1111.4723 [math.CO], 2011.
Sherry H. F. Yan and Yuexiao Xu, Self-dual interval orders and row-Fishburn matrices, Electronic Journal of Combinatorics, 19(2):P5 (2012).

Cf. A022493, A138265, A158690, A179525.

g:=sum(product(1-(1-x)^(2*i-1), i= 1..n), n = 0..20): gser:=series(g, x = 0,20): seq(coeff(gser, x, n), n = 0..19); # by definition
g:=sum((-1)^n*product(1-1/(1-x)^i, i= 1..n), n = 0..20): gser:=series(g, x = 0,20): seq(coeff(gser, x, n), n = 0..19);

a[ n_] := SeriesCoefficient[ Sum[ Product[ 1 - (1 - x)^(2 i - 1), {i, k}], {k, 0, n}], {x, 0, n}]; (* Michael Somos, Jun 27 2017 *)

{a(n)=polcoeff(sum(m=0, n, prod(k=1, m, 1/(1-x)^k-1, 1+x*O(x^n))), n)} /* Paul D. Hanna, Jan 29 2012 */

/* G.f. as a Continued Fraction: */
{a(n)=local(CF=1+x*O(x)); for(k=0, n, CF=1/(1 - (1-x)^(n-k+1)*(1-(1-x)^(n-k+2))*CF)); polcoeff(1/(1-x*CF), n, x)} /* Paul D. Hanna, Jan 29 2012 */

Sum_{n >= 0} Product_{i= 1..n} (1-(1-x)^(2*i-1)) = 1 + x + 3*x^2 + 12*x^3 + 61*x^4 + .... Compare with A022493, A138265 and A158690.
G.f.: Sum_{n>=0} Product_{k=1..n} [1/(1-x)^k - 1].
G.f.: 1/(1 - (1-(1-x))/(1 - (1-x)*(1-(1-x)^2)/(1 - (1-x)^2*(1-(1-x)^3)/(1 - (1-x)^3*(1-(1-x)^4)/(1 - (1-x)^4*(1-(1-x)^5)/(1 -...)))))), a continued fraction. - Paul D. Hanna, Jan 29 2012
By results of Bringmann, Li and Rhoades, a(n) is asymptotically c*n!*(12/Pi^2)^n, with c=6*sqrt(2)*exp(Pi^2/24)/Pi^2, and the ratio a(n)/A179525(n) tends to exp(Pi^2/12). - Vít Jelínek, Sep 04 2014
From Peter Bala, May 16 2017: (Start)
G.f. 1/2*( 1 + Sum_{n >= 0} 1/(1 - x)^((n+1)*(n+2)/2) * Product_{i = 1..n} (1 - (1 - x)^i) ).
Conjectural g.f.: Sum_{n >= 0} 1/(1 - x)^((n+1)*(2*n+1)) * Product_{i = 1..2*n} ((1 - x)^i - 1). (End)

A207386 G.f.: Sum_{n>=0} Product_{k=1..n} (q^k - 1) where q = (1+x)/(1+x^3).

Original entry on oeis.org

1, 1, 2, 6, 28, 172, 1269, 10879, 106343, 1167970, 14241792, 190919195, 2790920003, 44184957237, 753152722642, 13752229833566, 267809474619299, 5540559819166056, 121355678158129804, 2805498395990301867, 68265999939081386947, 1744058001878302097109

Offset: 0

Paul D. Hanna, Feb 17 2012

nonn

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.

			G.f.: A(x) = 1 + x + 2*x^2 + 6*x^3 + 28*x^4 + 172*x^5 + 1269*x^6 +...
Let q = (1+x)/(1+x^3) = 1/(1-x+x^2), then
A(x) = 1 + (q-1) + (q-1)*(q^2-1) + (q-1)*(q^2-1)*(q^3-1) + (q-1)*(q^2-1)*(q^3-1)*(q^4-1) + (q-1)*(q^2-1)*(q^3-1)*(q^4-1)*(q^5-1) +...
which also is proposed to equal:
A(x) = 1 + (q-1)/q + (q-1)*(q^3-1)/q^4 + (q-1)*(q^3-1)*(q^5-1)/q^9 + (q-1)*(q^3-1)*(q^5-1)*(q^7-1)/q^16 + (q-1)*(q^3-1)*(q^5-1)*(q^7-1)*(q^9-1)/q^25 +...

Vaclav Kotesovec, Table of n, a(n) for n = 0..250
Hsien-Kuei Hwang, Emma Yu Jin, Asymptotics and statistics on Fishburn matrices and their generalizations, arXiv:1911.06690 [math.CO], 2019.

Cf. A158690, A158691, A179525, A207397, A207433.

{a(n)=local(A=1+x,q=(1+x)/(1+x^3 +x*O(x^n))); A=sum(m=0,n,prod(k=1,m,(q^k-1)));polcoeff(A,n)}

{a(n)=local(A=1+x,q=(1+x)/(1+x^3 +x*O(x^n))); A=sum(m=0,n,q^(-m^2)*prod(k=1,m,(q^(2*k-1)-1)));polcoeff(A,n)}
for(n=0,21,print1(a(n),", "))

G.f.: Sum_{n>=0} q^(-n^2) * Product_{k=1..n} (q^(2*k-1) - 1) where q = (1+x)/(1+x^3). [Based on Peter Bala's conjecture in A158690]

a(n) ~ n! * 2^(2*n+3/2) * 3^(n+1) / (exp(Pi^2/24) * Pi^(2*n+2)). - Vaclav Kotesovec, Aug 22 2017

A207569 G.f.: Sum_{n>=0} Product_{k=1..n} ((1+x)^(2*k-1) - 1).

Original entry on oeis.org

1, 1, 3, 18, 151, 1640, 21825, 343763, 6253234, 128993019, 2975165831, 75866604098, 2119310099700, 64361149952242, 2111222815441491, 74391641880144734, 2802300974537717340, 112379709083552152423, 4780136025081921948194, 214954914688567198802759

Offset: 0

Paul D. Hanna, Feb 18 2012

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

			G.f.: A(x) = 1 + x + 3*x^2 + 18*x^3 + 151*x^4 + 1640*x^5 + 21825*x^6 + ...
such that, by definition,
A(x) = 1 + ((1+x)-1) + ((1+x)-1)*((1+x)^3-1) + ((1+x)-1)*((1+x)^3-1)*((1+x)^5-1) + ((1+x)-1)*((1+x)^3-1)*((1+x)^5-1)*((1+x)^7-1) + ...

Michael De Vlieger, Table of n, a(n) for n = 0..200
Hsien-Kuei Hwang and Emma Yu Jin, Asymptotics and statistics on Fishburn matrices and their generalizations, arXiv:1911.06690 [math.CO], 2019.

Cf. A179525, A207557, A207570, A207571, A215066, A053250, A158691.

CoefficientList[Series[Sum[Product[(1+x)^(2*k-1)-1, {k, 1, n}], {n, 0, 20}], {x, 0, 20}], x] (* Vaclav Kotesovec, May 06 2014 *)

{a(n)=polcoeff(sum(m=0,n,prod(k=1,m,(1+x)^(2*k-1)-1) +x*O(x^n)),n)}
for(n=0,25,print1(a(n),", "))

a(n) ~ sqrt(12) * 24^n * n^n / (exp(n+Pi^2/48) * Pi^(2*n+1)). - Vaclav Kotesovec, May 06 2014
G.f.: 1/2*( 1 + Sum_{n>=0} (1 + x)^(2*n+1) * Product_{k = 1..n} ((1 + x)^(2*k-1) - 1) ). Cf. A053250 and A215066. - Peter Bala, May 15 2017
Conjectural g.f.: Sum_{n>=0} (-1)^n*Product_{k = 1..n} 1 + ( -1/(1 + x) )^k. - Peter Bala, Dec 04 2020
From Peter Bala, Jan 29 2021: (Start)
Conjectural g.f.s: Sum_{n >= 0} (-1)^n*(1 + x)^(n+1)*Product_{k = 1..n} (1 + (-1)^k*(1 + x)^k)^2. Also
(1/2)*( 1 + Sum_{n >= 0} 1/(1 + x)^(n+1)*Product_{k = 1..n} (1 + (-1)^k/(1 + x)^k) ). (End)

A207397 G.f.: Sum_{n>=0} Product_{k=1..n} (q^k - 1) where q = (1+x)/(1+x^2).

Original entry on oeis.org

1, 1, 1, 2, 11, 74, 557, 4799, 47004, 516717, 6302993, 84502346, 1235198136, 19552296646, 333212892221, 6083009119262, 118433569748072, 2449663066933397, 53643715882853914, 1239875630317731463, 30163779836127304106, 770476745704778418686

Offset: 0

Paul D. Hanna, Feb 17 2012

nonn

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.

			G.f.: A(x) = 1 + x + x^2 + 2*x^3 + 11*x^4 + 74*x^5 + 557*x^6 + 4799*x^7 +...
Let q = (1+x)/(1+x^2), then
A(x) = 1 + (q-1) + (q-1)*(q^2-1) + (q-1)*(q^2-1)*(q^3-1) + (q-1)*(q^2-1)*(q^3-1)*(q^4-1) + (q-1)*(q^2-1)*(q^3-1)*(q^4-1)*(q^5-1) +...
which also is proposed to equal:
A(x) = 1 + (q-1)/q + (q-1)*(q^3-1)/q^4 + (q-1)*(q^3-1)*(q^5-1)/q^9 + (q-1)*(q^3-1)*(q^5-1)*(q^7-1)/q^16 + (q-1)*(q^3-1)*(q^5-1)*(q^7-1)*(q^9-1)/q^25 +...

Vaclav Kotesovec, Table of n, a(n) for n = 0..240
Hsien-Kuei Hwang, Emma Yu Jin, Asymptotics and statistics on Fishburn matrices and their generalizations, arXiv:1911.06690 [math.CO], 2019.

Cf. A158690, A158691, A179525, A207386, A207433.

{a(n)=local(A=1+x,q=(1+x)/(1+x^2 +x*O(x^n))); A=sum(m=0,n,prod(k=1,m,(q^k-1)));polcoeff(A,n)}

{a(n)=local(A=1+x,q=(1+x)/(1+x^2 +x*O(x^n))); A=sum(m=0,n,q^(-m^2)*prod(k=1,m,(q^(2*k-1)-1)));polcoeff(A,n)}
for(n=0,25,print1(a(n),", "))

G.f.: Sum_{n>=0} q^(-n^2) * Product_{k=1..n} (q^(2*k-1) - 1) where q = (1+x)/(1+x^2). [Based on Peter Bala's conjecture in A158690]

a(n) ~ c * 12^n * n! / Pi^(2*n), where c = 6*sqrt(2) / (Pi^2 * exp(Pi^2/8)) = 0.250367043877216848533826021231826... . - Vaclav Kotesovec, May 06 2014, updated Aug 22 2017

A207433 G.f.: Sum_{n>=0} Product_{k=1..n} (q^k - 1) where q = (1-x^3)/(1-x).

Original entry on oeis.org

1, 1, 3, 11, 56, 350, 2609, 22582, 222625, 2462969, 30219676, 407276420, 5981197376, 95073427910, 1626294895274, 29788176027819, 581704672430937, 12064521684969823, 264843222932272690, 6135057298705027024, 149559103545555671423, 3827360866024134614644

Offset: 0

Paul D. Hanna, Feb 17 2012

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

			G.f.: A(x) = 1 + x + 3*x^2 + 11*x^3 + 56*x^4 + 350*x^5 + 2609*x^6 +...
Let q = (1-x^3)/(1-x) = 1 + x + x^2, then
A(x) = 1 + (q-1) + (q-1)*(q^2-1) + (q-1)*(q^2-1)*(q^3-1) + (q-1)*(q^2-1)*(q^3-1)*(q^4-1) + (q-1)*(q^2-1)*(q^3-1)*(q^4-1)*(q^5-1) +...
Also, we have the identity:
A(x) = 1 + (q-1)/q + (q-1)*(q^3-1)/q^4 + (q-1)*(q^3-1)*(q^5-1)/q^9 + (q-1)*(q^3-1)*(q^5-1)*(q^7-1)/q^16 + (q-1)*(q^3-1)*(q^5-1)*(q^7-1)*(q^9-1)/q^25 +...
From _Peter Bala_, Nov 05 2017: (Start)
a(3) = 11: The eleven row Fishburn matrices of size 3 with entries in {0,1,2} are
/1 0\ /2 0\ /0 1\ /0 2\ /1 1\
\0 2/ \0 1/ \0 2/ \0 1/ \0 1/
and
/1 0 0\ /0 1 0\ /0 0 1\ /1 0 0\ /0 1 0\ /0 0 1\
|0 1 0| |0 1 0| |0 0 1| |0 0 1| |0 0 1| |0 0 1|.
\0 0 1/ \0 0 1/ \0 0 1/ \0 0 1/ \0 0 1/ \0 0 1/
(End)

Vaclav Kotesovec, Table of n, a(n) for n = 0..260
Hsien-Kuei Hwang, Emma Yu Jin, Asymptotics and statistics on Fishburn matrices and their generalizations, arXiv:1911.06690 [math.CO], 2019.

Cf. A158690, A158691, A179525, A207386, A207397.

{a(n)=local(A=1+x,q=(1+x+x^2 +x*O(x^n))); A=sum(m=0,n,prod(k=1,m,(q^k-1)));polcoeff(A,n)}

{a(n)=local(A=1+x,q=(1+x+x^2 +x*O(x^n))); A=sum(m=0,n,q^(-m^2)*prod(k=1,m,(q^(2*k-1)-1)));polcoeff(A,n)}
for(n=0,25,print1(a(n),", "))

G.f.: Sum_{n>=0} q^(-n^2) * Product_{k=1..n} (q^(2*k-1) - 1) where q = (1-x^3)/(1-x). [Based on Peter Bala's conjecture in A158690]

a(n) ~ exp(Pi^2/24) * 2^(2*n+3/2) * 3^(n+1) * n! / Pi^(2*n+2). - Vaclav Kotesovec, Aug 22 2017

A207556 G.f.: Sum_{n>=0} (1+x)^n * Product_{k=1..n} ((1+x)^k - 1).

Original entry on oeis.org

1, 1, 3, 11, 55, 339, 2499, 21433, 209717, 2305719, 28141925, 377579731, 5523750291, 87508680045, 1492510215135, 27266981038343, 531245913925837, 10995334516297279, 240925208376757203, 5571653169126500083, 135617881389268715939, 3465772763274106884733

Offset: 0

Paul D. Hanna, Feb 18 2012

nonn

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

			G.f.: A(x) = 1 + x + 3*x^2 + 11*x^3 + 55*x^4 + 339*x^5 + 2499*x^6 +...
such that, by definition,
A(x) = 1 + (1+x)*((1+x)-1) + (1+x)^2*((1+x)-1)*((1+x)^2-1) + (1+x)^3*((1+x)-1)*((1+x)^2-1)*((1+x)^3-1) + (1+x)^4*((1+x)-1)*((1+x)^2-1)*((1+x)^3-1)*((1+x)^4-1) +...

Vaclav Kotesovec, Table of n, a(n) for n = 0..170
Hsien-Kuei Hwang, Emma Yu Jin, Asymptotics and statistics on Fishburn matrices and their generalizations, arXiv:1911.06690 [math.CO], 2019.

Cf. A179525, A207557.

{a(n)=polcoeff(sum(m=0,n,(1+x)^m*prod(k=1,m,(1+x)^k-1) +x*O(x^n)),n)}
for(n=0,25,print1(a(n),", "))

a(n) ~ 2 * 12^(n+1) * n^(n+1/2) / (exp(n+Pi^2/24) * Pi^(2*n+3/2)). - Vaclav Kotesovec, May 07 2014

A289313 The number of upper-triangular matrices with integer entries whose absolute sum is equal to n and such that each row contains a nonzero entry.

Original entry on oeis.org

1, 2, 10, 74, 722, 8786, 128218, 2182554, 42456226, 929093538, 22590839466, 604225121258, 17630145814898, 557285515817970, 18970857530674554, 691929648113663802, 26919562120779248962, 1112769248605003393858, 48704349211392743606602

Offset: 0

Peter Bala, Jul 02 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(2) = 10: The ten generalized row-Fishburn matrices of size 2 are
  (+-2),
  /+-1  0\ and  /0 +-1\
  |      |      |     |
  \0  +-1/      \0 +-1/.

I. P. Goulden and D. M. Jackson, Combinatorial Enumeration, Wiley, N.Y., 1983, p. 42.

Seiichi Manyama, Table of n, a(n) for n = 0..200
Hsien-Kuei Hwang and Emma Yu Jin, Asymptotics and statistics on Fishburn matrices and their generalizations, arXiv:1911.06690 [math.CO], 2019.

Cf. A022493, A158691, A179525, A289312.

G:= add(mul( ((1 + x)/(1 - x))^i - 1, i=1..n),n=0..20):
S:= series(G,x,21):
seq(coeff(S,x,j),j=0..20);
# Peter Bala, Jul 24 2017

O.g.f.: Sum_{n >= 0} ( Product_{i = 1..n} ((1 + x)/(1 - x))^i - 1 ).
The o.g.f. has several alternative forms:
Sum_{n >= 0} ( Product_{i = 1..n} ( 1 - ((1 - x)/(1 + x))^(2*i-1) ) );
Sum_{n >= 0} ((1 - x)/(1 + x))^(n+1) * ( Product_{i = 1..n} 1 - ((1 - x)/(1 + x))^(2*i) );
1/2*( 1 + Sum_{n >= 0} ((1 + x)/(1 - x))^((n+1)*(n+2)/2) * Product_{i = 1..n} ( 1 - ((1 - x)/(1 + x))^i ) ).
Conjectural g.f.: Sum_{n >= 0} ((1 + x)/(1 - x))^((n+1)*(2*n+1)) * Product_{i = 1..2*n} ( ((1 - x)/(1 + x))^i - 1 ).
a(n) ~ 2^(3*n+2) * 3^(n+1) * n^(n + 1/2) / (exp(n) * Pi^(2*n + 3/2)). - Vaclav Kotesovec, Aug 31 2023

A207557 G.f.: Sum_{n>=0} 1/(1+x)^(n^2-n) * Product_{k=1..n} ((1+x)^(2*k-1) - 1).

Original entry on oeis.org

1, 1, 3, 12, 64, 420, 3276, 29581, 303389, 3483053, 44245695, 616103046, 9330961666, 152700926414, 2685132170466, 50488787588936, 1010864433071206, 21470488933116138, 482176661100286182, 11415700804801064258, 284169548252819022230, 7419733139418740010570

Offset: 0

Paul D. Hanna, Feb 18 2012

nonn

Compare g.f. to: Sum_{n>=0} 1/(1+x)^(n^2) * Product_{k=1..n} ((1+x)^(2*k-1) - 1), which is the g.f. of A179525.

			G.f.: A(x) = 1 + x + 3*x^2 + 12*x^3 + 64*x^4 + 420*x^5 + 3276*x^6 +...
such that, by definition,
A(x) = 1 + ((1+x)-1) + ((1+x)-1)*((1+x)^3-1)/(1+x)^2 + ((1+x)-1)*((1+x)^3-1)*((1+x)^5-1)/(1+x)^6 + ((1+x)-1)*((1+x)^3-1)*((1+x)^5-1)*((1+x)^7-1)/(1+x)^20 +...

Vaclav Kotesovec, Table of n, a(n) for n = 0..135
Hsien-Kuei Hwang, and Emma Yu Jin, Asymptotics and statistics on Fishburn matrices and their generalizations, arXiv:1911.06690 [math.CO], p. 36, 2019.

Cf. A179525, A207556.

{a(n)=polcoeff(sum(m=0,n,prod(k=1,m,(1+x)^(2*k-1)-1)/(1+x+x*O(x^n))^(m^2-m) ),n)}
for(n=0,25,print1(a(n),", "))

Given A(x) is the g.f. of this sequence, note that:
1 + x*A(x) = Sum_{n>=0} 1/(1+x)^(n^2+n) * Product_{k=1..n} ((1+x)^(2*k-1) - 1).
a(n) ~ 2*sqrt(6) * 12^(n+1) * n^(n+1) / (exp(n+Pi^2/24) * Pi^(2*n+3)). - Vaclav Kotesovec, May 07 2014

A207570 G.f.: Sum_{n>=0} Product_{k=1..n} ((1+x)^(3*k-2) - 1).

Original entry on oeis.org

1, 1, 4, 34, 410, 6455, 125251, 2888305, 77157780, 2342972405, 79701049425, 3002132647515, 124039845584382, 5577660227565634, 271162541308698623, 14172237715785139175, 792418822364402364530, 47198077739119663907870, 2983413619934353599892285

Offset: 0

Paul D. Hanna, Feb 18 2012

nonn

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

			G.f.: A(x) = 1 + x + 4*x^2 + 34*x^3 + 410*x^4 + 6455*x^5 + 125251*x^6 +...
such that, by definition,
A(x) = 1 + ((1+x)-1) + ((1+x)-1)*((1+x)^4-1) + ((1+x)-1)*((1+x)^4-1)*((1+x)^7-1) + ((1+x)-1)*((1+x)^4-1)*((1+x)^7-1)*((1+x)^10-1) +...

Michael De Vlieger, Table of n, a(n) for n = 0..200
Hsien-Kuei Hwang, Emma Yu Jin, Asymptotics and statistics on Fishburn matrices and their generalizations, arXiv:1911.06690 [math.CO], 2019.

Cf. A179525, A207556, A207569, A207571.

Join[{1},Rest[With[{nn=20},CoefficientList[Series[Sum[Product[ (1+x)^(3k-2)-1,{k,n}],{n,nn}],{x,0,nn}],x]]]] (* Harvey P. Dale, Aug 20 2012 *)

{a(n)=polcoeff(sum(m=0,n,prod(k=1,m,(1+x)^(3*k-2)-1) +x*O(x^n)),n)}
for(n=0,25,print1(a(n),", "))

a(n) ~ GAMMA(2/3) * 2^(2*n-1/3) * 3^(2*n+5/6) * n^(n-1/6) / (exp(n+Pi^2/72) * Pi^(2*n+7/6)). - Vaclav Kotesovec, May 06 2014

A207571 G.f.: Sum_{n>=0} Product_{k=1..n} ((1+x)^(3*k-1) - 1).

Original entry on oeis.org

1, 2, 11, 105, 1390, 23520, 484247, 11742927, 327711230, 10343198878, 364237027076, 14156867852699, 601927703437645, 27790427952836499, 1384496764982434033, 74027620787319243688, 4228343290201028904807, 256946673653717460509502, 16551666142815138743519611

Offset: 0

Paul D. Hanna, Feb 18 2012

nonn

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

			G.f.: A(x) = 1 + 2*x + 11*x^2 + 105*x^3 + 1390*x^4 + 23520*x^5 + 484247*x^6 +...
such that, by definition,
A(x) = 1 + ((1+x)^2-1) + ((1+x)^2-1)*((1+x)^5-1) + ((1+x)^2-1)*((1+x)^5-1)*((1+x)^8-1) + ((1+x)^2-1)*((1+x)^5-1)*((1+x)^8-1)*((1+x)^11-1) +...

Michael De Vlieger, Table of n, a(n) for n = 0..200
Hsien-Kuei Hwang, Emma Yu Jin, Asymptotics and statistics on Fishburn matrices and their generalizations, arXiv:1911.06690 [math.CO], 2019.

Cf. A179525, A207556, A207569, A207570.

CoefficientList[Series[Sum[Product[(1+x)^(3*k-1)-1, {k, 1, n}], {n, 0, 20}], {x, 0, 20}], x] (* Vaclav Kotesovec, May 06 2014 *)

{a(n)=polcoeff(sum(m=0,n,prod(k=1,m,(1+x)^(3*k-1)-1) +x*O(x^n)),n)}
for(n=0,25,print1(a(n),", "))

a(n) ~ GAMMA(1/3) * 2^(2*n+1/3) * 3^(2*n+7/6) * n^(n+1/6) / (exp(n+Pi^2/72) * Pi^(2*n+11/6)). - Vaclav Kotesovec, May 06 2014

cp's OEIS Frontend

A158691 The number of upper-triangular matrices with at least one nonzero entry in each row and whose entries sum to n.

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A207386 G.f.: Sum_{n>=0} Product_{k=1..n} (q^k - 1) where q = (1+x)/(1+x^3).

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A207569 G.f.: Sum_{n>=0} Product_{k=1..n} ((1+x)^(2*k-1) - 1).

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A207397 G.f.: Sum_{n>=0} Product_{k=1..n} (q^k - 1) where q = (1+x)/(1+x^2).

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A207433 G.f.: Sum_{n>=0} Product_{k=1..n} (q^k - 1) where q = (1-x^3)/(1-x).

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A207556 G.f.: Sum_{n>=0} (1+x)^n * Product_{k=1..n} ((1+x)^k - 1).

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A289313 The number of upper-triangular matrices with integer entries whose absolute sum is equal to n and such that each row contains a nonzero entry.

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