This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A158691 #89 Aug 22 2023 06:36:34
%S A158691 1,1,3,12,61,380,2815,24213,237348,2612681,31915787,428481472,
%T A158691 6271362282,99388642292,1695614865711,30984649882928,603790447393402,
%U A158691 12498732438500663,273902239550757626,6334968666307580051,154211723833861061644,3941258052200287007636
%N A158691 The number of upper-triangular matrices with at least one nonzero entry in each row and whose entries sum to n.
%C A158691 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.
%C A158691 From _Vít Jelínek_, Sep 04 2014: (Start)
%C A158691 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:
%C A158691 F(x) = Sum_{n>=0} Product_{i=1..n} (1-(1-x)^(2*i-1))
%C A158691 F(x) = Sum_{n>=0} Product_{i=1..n} (1/(1-x)^i - 1)
%C A158691 F(x) = Sum_{n>=0} (1-x)^(n+1) Product_{i=1..n} (1-(1-x)^(2*i))
%C A158691 The first two formulas were conjectured to be equal by Bala. This was confirmed by Andrews and Jelínek, who also derived the third formula.
%C A158691 Bringmann, Li and Rhoades have independently derived the three formulas above, and additionaly, they proved that
%C A158691 F(x) = (1/2)*Sum_{n>=0} Product_{i=1..n} (1/(1+(1-x)^i)).
%C A158691 (End)
%C A158691 From _Vít Jelínek_, Feb 12 2012: (Start)
%C A158691 a(n) has the following combinatorial interpretations:
%C A158691 (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):
%C A158691 (2), (1.) and (.1)
%C A158691 (.1) (.1)
%C A158691 (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:
%C A158691 (2.), (11.) and (1...)
%C A158691 (.2) (..1) (.1..)
%C A158691 (..1) (..1.)
%C A158691 (...1)
%C A158691 (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:
%C A158691 (2), (11) and (1..)
%C A158691 (.1) (.1.)
%C A158691 (..1)
%C A158691 (End)
%H A158691 Seiichi Manyama, <a href="/A158691/b158691.txt">Table of n, a(n) for n = 0..200</a>
%H A158691 George E. Andrews, Vít Jelínek, <a href="http://arxiv.org/abs/1309.6669">On q-Series Identities Related to Interval Orders</a>, arXiv:1309.6669 [math.CO], 2013.
%H A158691 George E. Andrews, Vít Jelínek, <a href="http://dx.doi.org/10.1016/j.ejc.2014.01.003">On q-Series Identities Related to Interval Orders</a>, European Journal of Combinatorics, Volume 39, July 2014, 178-187.
%H A158691 Kathrin Bringmann, Yingkun Li, Robert C. Rhoades, <a href="http://dx.doi.org/10.1016/j.ejc.2014.04.003">Asymptotics for the number of row-Fishburn matrices</a>, European Journal of Combinatorics, Volume 41, October 2014, Pages 183-196; <a href="http://www.mi.uni-koeln.de/Bringmann/selfdual_rev7.pdf">preprint</a>.
%H A158691 Hsien-Kuei Hwang, Emma Yu Jin, <a href="https://arxiv.org/abs/1911.06690">Asymptotics and statistics on Fishburn matrices and their generalizations</a>, arXiv:1911.06690 [math.CO], 2019.
%H A158691 Florian Ingels, Romain Azaïs, <a href="https://arxiv.org/abs/2003.08144">Enumeration of Unordered Forests</a>, arXiv:2003.08144 [cs.DM], 2020.
%H A158691 Vít Jelínek, <a href="http://arxiv.org/abs/1106.2261">Counting self-dual interval orders</a>, arXiv:1106.2261 [math.CO], 2011.
%H A158691 Vít Jelínek, <a href="http://dx.doi.org/10.1016/j.jcta.2011.11.010">Counting general and self-dual interval orders</a>, Journal of Combinatorial Theory, Series A, Volume 119, Issue 3, April 2012, pp. 599-614.
%H A158691 Sherry H. F. Yan and Yuexiao Xu, <a href="http://arxiv.org/abs/1111.4723">Self-dual interval orders and row-Fishburn matrices</a>, arXiv:1111.4723 [math.CO], 2011.
%H A158691 Sherry H. F. Yan and Yuexiao Xu, <a href="http://www.combinatorics.org/ojs/index.php/eljc/article/view/v19i2p5">Self-dual interval orders and row-Fishburn matrices</a>, Electronic Journal of Combinatorics, 19(2):P5 (2012).
%F A158691 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.
%F A158691 G.f.: Sum_{n>=0} Product_{k=1..n} [1/(1-x)^k - 1].
%F A158691 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
%F A158691 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
%F A158691 From _Peter Bala_, May 16 2017: (Start)
%F A158691 G.f. 1/2*( 1 + Sum_{n >= 0} 1/(1 - x)^((n+1)*(n+2)/2) * Product_{i = 1..n} (1 - (1 - x)^i) ).
%F A158691 Conjectural g.f.: Sum_{n >= 0} 1/(1 - x)^((n+1)*(2*n+1)) * Product_{i = 1..2*n} ((1 - x)^i - 1). (End)
%e A158691 G.f.: A(x) = 1 + x + 3*x^2 + 12*x^3 + 61*x^4 + 380*x^5 + 2815*x^6 +...
%p A158691 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
%p A158691 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);
%t A158691 a[ n_] := SeriesCoefficient[ Sum[ Product[ 1 - (1 - x)^(2 i - 1), {i, k}], {k, 0, n}], {x, 0, n}]; (* _Michael Somos_, Jun 27 2017 *)
%o A158691 (PARI) {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 */
%o A158691 (PARI) /* G.f. as a Continued Fraction: */
%o A158691 {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 */
%Y A158691 Cf. A022493, A138265, A158690, A179525.
%K A158691 easy,nonn
%O A158691 0,3
%A A158691 _Peter Bala_, Mar 24 2009