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

 

Logo
Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A005321 Upper triangular n X n (0,1)-matrices with no zero rows or columns.
(Formerly M1986)
17
1, 1, 2, 10, 122, 3346, 196082, 23869210, 5939193962, 2992674197026, 3037348468846562, 6189980791404487210, 25285903982959247885402, 206838285372171652078912306, 3386147595754801373061066905042, 110909859519858523995273393471390010 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,3
REFERENCES
T. L. Greenough, Enumeration of interval orders without duplicated holdings, Preprint, circa 1976.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
LINKS
E. Andresen and K. Kjeldsen, On certain subgraphs of a complete transitively directed graph, Discrete Math. 14 (1976), no. 2, 103-119.
T. Lockman Greenough, Representation and enumeration of interval orders and semiorders, Ph.D. Thesis, Dartmouth, 1976.
T. L. Greenough, Enumeration of interval orders without duplicated holdings, Preprint, circa 1976. [Annotated scanned copy]
T. Lockman Greenough, Enumeration of interval orders without duplicated holdings, in Notices of the AMS, February 1976, page A-314.
T. L. Greenough and K. P. Bogart, The Representation and Enumeration of Interval Orders, Preprint, circa 1976. [Annotated scanned copy]
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 Self-Dual Interval Orders, arXiv:1106.2261 [math.CO], 2011. See Corollary 2.4.
Vít Jelínek, Counting general and self-dual interval orders, Journal of Combinatorial Theory, Series A, Volume 119, Issue 3, April 2012, pp. 599-614. See Corollary 2.4.
J. Longyear, T. Trotter, N. J. A. Sloane, Correspondence
FORMULA
a(n) = Sum_{k=0..n} binomial(n,k)*A005327(k+1).
G.f.: Sum_{n >= 0} x^n*Product_{i = 1..n} ((2^i-1)/(1 + (2^i-1)*x)). - Vladeta Jovovic, Mar 10 2008
From Peter Bala, Jul 06 2017: (Start)
Two conjectural continued fractions for the o.g.f.:
1/(1 - x/(1 - x/(1 - 6*x/(1 - 9*x/(1 - 28*x/(1 - 49*x/(1 - ... - 2^(n-1)*(2^n - 1)*x/(1 - (2^n - 1)^2*x/(1 - ...)))))))));
1 + x/(1 - 2*x/(1 - 3*x/(1 - 12*x/(1 - 21*x/(1 - ... - 2^n*(2^n - 1)*x/(1 - (2^(n+1) - 1)*(2^n - 1)*x/(1 - ...))))))). Cf. A289314 and A289315. (End)
a(n) = (-1)^n*Sum_{k=0..n} qS2(n,k)*[k]!*(-1)^k, where qS2(n,k) is the triangle A139382, and [k]! is q-factorial, q=2. - Vladimir Kruchinin, Oct 10 2019
a(n) = 1 + Sum_{k=2..n} binomial(n,k)*Sum{i=2..k} (-1)^i*Product_{j=i+1..k} (2^j - 1). See Greenough. - Michel Marcus, Oct 13 2019
MATHEMATICA
max = 14; f[x_] := Sum[ x^n*Product[ (2^i-1) / (1+(2^i-1)*x), {i, 1, n}], {n, 0, max}]; CoefficientList[ Series[ f[x], {x, 0, max}], x] (* Jean-François Alcover, Nov 23 2011, after Vladeta Jovovic *)
PROG
(PARI) a(n) = 1 + sum(k=2, n, binomial(n, k)*sum(i=2, k, (-1)^i*prod(j=i+1, k, 2^j - 1))); \\ Michel Marcus, Oct 13 2019
CROSSREFS
Column sums of A137252.
Sequence in context: A256832 A060690 A013038 * A339934 A348876 A092645
KEYWORD
nonn,easy,nice
AUTHOR
EXTENSIONS
More terms from Max Alekseyev, Apr 27 2010
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.

License Agreements, Terms of Use, Privacy Policy. .

Last modified March 18 22:56 EDT 2024. Contains 370952 sequences. (Running on oeis4.)