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A005321 Upper triangular n X n (0,1)-matrices with no zero rows or columns.
(Formerly M1986)
15
1, 1, 2, 10, 122, 3346, 196082, 23869210, 5939193962, 2992674197026, 3037348468846562, 6189980791404487210, 25285903982959247885402, 206838285372171652078912306, 3386147595754801373061066905042 (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

Vincenzo Librandi, Table of n, a(n) for n = 0..80

E. Andresen, 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, K. P. Bogart, The Representation and Enumeration of Interval Orders, Preprint, circa 1976. [Annotated scanned copy]

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

Index entries for sequences related to binary matrices

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

Cf. A022493, A138265, A289314, A289315.

Sequence in context: A256832 A060690 A013038 * A092645 A202950 A144835

Adjacent sequences:  A005318 A005319 A005320 * A005322 A005323 A005324

KEYWORD

nonn,easy,nice

AUTHOR

N. J. A. Sloane

EXTENSIONS

More terms from Max Alekseyev, Apr 27 2010

STATUS

approved

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Last modified November 12 04:21 EST 2019. Contains 329051 sequences. (Running on oeis4.)