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A005321
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Upper triangular n X n (0,1)-matrices with no zero rows or columns.
(Formerly M1986)
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15
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1, 1, 2, 10, 122, 3346, 196082, 23869210, 5939193962, 2992674197026, 3037348468846562, 6189980791404487210, 25285903982959247885402, 206838285372171652078912306, 3386147595754801373061066905042
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OFFSET
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0,3
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REFERENCES
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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).
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LINKS
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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
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FORMULA
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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
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MATHEMATICA
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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 *)
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PROG
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(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
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CROSSREFS
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Cf. A022493, A138265, A289314, A289315.
Sequence in context: A256832 A060690 A013038 * A339934 A092645 A333455
Adjacent sequences: A005318 A005319 A005320 * A005322 A005323 A005324
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KEYWORD
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nonn,easy,nice
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AUTHOR
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N. J. A. Sloane
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EXTENSIONS
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More terms from Max Alekseyev, Apr 27 2010
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STATUS
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approved
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