The OEIS is supported by the many generous donors to the OEIS Foundation.

 Hints (Greetings from The On-Line Encyclopedia of Integer Sequences!)
 A289315 Number of n X n Fishburn matrices with entries in the set {0,1,2,3}. 5
 1, 3, 36, 2052, 505764, 511718148, 2088275673636, 34176650317115652, 2239082850356711468964, 586908388119824949146284548, 615402202729113953115253336166436, 2581165458211746544190089033131172341252, 43304685245392697816407075492352986194550240164 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 COMMENTS 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. LINKS Robert Israel, Table of n, a(n) for n = 0..57 Hsien-Kuei Hwang, Emma Yu Jin, and Michael J. Schlosser, Asymptotics and statistics on Fishburn Matrices: dimension distribution and a conjecture of Stoimenow, arXiv:2012.13570 [math.CO], 2020. Vít Jelínek, Counting general and self-dual interval orders, Journal of Combinatorial Theory, Series A, Volume 119, Issue 3, April 2012, pp. 599-614; arXiv preprint, arXiv:1106.2261 [math.CO], 2011. FORMULA O.g.f.: A(x) = Sum_{n >= 0} x^n Product_{i = 1..n} (4^i - 1)/(1 + x*(4^i - 1)) = 1 + 3*x + 36*x^2 + 2052*x^3 + ... (use Jelínek, Theorem 2.1 with v = w = x = y = 3). Two conjectural continued fractions for the o.g.f.: A(x) = 1/(1 - 3*x/(1 - 9*x/(1 - 60*x/(1 - 225*x/(1 - 1008*x/(1 - 3969*x/(1 - ... - 4^(n-1)*(4^n - 1)*x/(1 - (4^n - 1)^2*x/(1 - ...))))))))) and A(x) = 1 + 3*x/(1 - 12*x/(1 - 45*x/(1 - 240*x/(1 - 945*x/(1 - ... - 4^n*(4^n - 1)*x/(1 - (4^(n+1) - 1)*(4^n - 1)*x/(1 - ...))))))). EXAMPLE a(2) = 36: The 36 2 X 2 Fishburn matrices with entries 0, 1, 2 or 3 are /1 0\ /1 0\ /2 0\ /2 0\ \0 1/ \0 2/ \0 1/ \0 2/ /1 0\ /3 0\ /3 0\ \0 3/ \0 1/ \0 3/ /2 0\ /3 0\ \0 3/ \0 2/ /1 2\ /1 1\ /1 2\ /2 1\ /2 2\ /2 1\ /2 2\ \0 1/ \0 2/ \0 2/ \0 1/ \0 1/ \0 2/ \0 2/ /1 1\ /1 3\ /1 1\ /1 3\ /3 1\ /3 3\ /3 1\ \0 1/ \0 1/ \0 3/ \0 3/ \0 1/ \0 1/ \0 3/ /2 3\ /2 2\ /2 3\ /3 2\ /3 3\ /3 2\ /3 3\ \0 2/ \0 3/ \0 3/ \0 2/ \0 2/ \0 3/ \0 3/ /1 2\ /1 3\ /2 3\ /2 1\ /3 1\ /3 2\. \0 3/ \0 2/ \0 1/ \0 3/ \0 2/ \0 1/ Alternatively, the 36 3-colored primitive Fishburn matrices of dimension 2 (using 1_j, j = 1,2,3 to denote the three different colored versions of 1) are /1_j  0\ (9 possibilities) \ 0 1_j/ and /1_j 1_j\ (27 possibilities). \ 0  1_j/ MAPLE G:= add(x^n*mul((4^i-1)/(1+x*(4^i-1)), i=1..n), n=0..40): S:= series(G, x, 41): seq(coeff(S, x, j), j=0..40); # Robert Israel, Jul 10 2017 MATHEMATICA Table[SeriesCoefficient[Sum[x^j*Product[(4^i - 1)/(1 + x (4^i - 1)), {i, j}], {j, 0, n}], {x, 0, n}], {n, 0, 12}] (* Michael De Vlieger, Jul 10 2017, after Maple by Robert Israel *) CROSSREFS Cf. A005321, A022493, A138265, A289314. Sequence in context: A300770 A272660 A193302 * A102579 A136393 A168370 Adjacent sequences:  A289312 A289313 A289314 * A289316 A289317 A289318 KEYWORD nonn,easy AUTHOR Peter Bala, Jul 03 2017 STATUS approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recents
The OEIS Community | Maintained by The OEIS Foundation Inc.

Last modified August 7 23:50 EDT 2022. Contains 355995 sequences. (Running on oeis4.)