A081064 -id:A081064 - cp's OEIS frontend

A003024 Number of acyclic digraphs (or DAGs) with n labeled nodes.

1, 1, 3, 25, 543, 29281, 3781503, 1138779265, 783702329343, 1213442454842881, 4175098976430598143, 31603459396418917607425, 521939651343829405020504063, 18676600744432035186664816926721, 1439428141044398334941790719839535103, 237725265553410354992180218286376719253505

Offset: 0

N. J. A. Sloane

Also the number of n X n real (0,1)-matrices with all eigenvalues positive. - Conjectured by Eric W. Weisstein, Jul 10 2003 and proved by McKay et al. 2003, 2004
Also the number of n X n real (0,1)-matrices with permanent equal to 1, up to permutation of rows/columns, cf. A089482. - Vladeta Jovovic, Oct 28 2009
Also the number of nilpotent elements in the semigroup of binary relations on [n]. - Geoffrey Critzer, May 26 2022
From Gus Wiseman, Jan 01 2024: (Start)
Also the number of sets of n nonempty subsets of {1..n} such that there is a unique way to choose a different element from each. For example, non-isomorphic representatives of the a(3) = 25 set-systems are:
  {{1},{2},{3}}
  {{1},{2},{1,3}}
  {{1},{2},{1,2,3}}
  {{1},{1,2},{1,3}}
  {{1},{1,2},{2,3}}
  {{1},{1,2},{1,2,3}}
These set-systems have ranks A367908, subset of A367906, for multisets A368101.
The version for no ways is A368600, any length A367903, ranks A367907.
The version for at least one way is A368601, any length A367902.
(End)

			For n = 2 the three (0,1)-matrices are {{{1, 0}, {0, 1}}, {{1, 0}, {1, 1}}, {{1, 1}, {0, 1}}}.

Archer, K., Gessel, I. M., Graves, C., & Liang, X. (2020). Counting acyclic and strong digraphs by descents. Discrete Mathematics, 343(11), 112041.
S. R. Finch, Mathematical Constants, Cambridge, 2003, p. 310.
F. Harary and E. M. Palmer, Graphical Enumeration, Academic Press, NY, 1973, p. 19, Eq. (1.6.1).
R. W. Robinson, Counting labeled acyclic digraphs, pp. 239-273 of F. Harary, editor, New Directions in the Theory of Graphs. Academic Press, NY, 1973.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
R. P Stanley, Enumerative Combinatorics I, 2nd. ed., p. 322.

Alois P. Heinz, Table of n, a(n) for n = 0..77 (first 41 terms from T. D. Noe)
T. E. Allen, J. Goldsmith, and N. Mattei, Counting, Ranking, and Randomly Generating CP-nets, 2014.
Huantian Cao, AutoGF: An Automated System to Calculate Coefficients of Generating Functions, thesis, 2002.
Huantian Cao, AutoGF: An Automated System to Calculate Coefficients of Generating Functions, thesis, 2002 [Local copy, with permission]
Eunice Y.-J. Chen, A. Choi, and A. Darwiche, On Pruning with the MDL Score, JMLR: Workshop and Conference Proceedings vol 52, 98-109, 2016.
S. Engstrom, Random acyclic orientations of graphs, Master's thesis written at the department of Mathematics at the Royal Institute of Technology (KTH) in Stockholm, Jan. 2013.
Zehao Jin, Mario Pasquato, Benjamin L. Davis, Tristan Deleu, Yu Luo, Changhyun Cho, Pablo Lemos, Laurence Perreault-Levasseur, Yoshua Bengio, Xi Kang, Andrea Valerio Maccio, and Yashar Hezaveh, A Data-driven Discovery of the Causal Connection between Galaxy and Black Hole Evolution, arXiv:2410.00965 [astro-ph.GA], 2024. See p. 33.
Zehao Jin, Mario Pasquato, Benjamin L. Davis, Andrea V. Macciò, and Yashar Hezaveh, Beyond Causal Discovery for Astronomy: Learning Meaningful Representations with Independent Component Analysis, arXiv:2410.14775 [astro-ph.GA], 2024. See pp. 2, 4, 10.
Zehao Jin, Mario Pasquato, Benjamin L. Davis, Tristan Deleu, Yu Luo, Changhyun Cho, Pablo Lemos, Laurence Perreault-Levasseur, Yoshua Bengio, and Xi Kang, Causal Discovery in Astrophysics: Unraveling Supermassive Black Hole and Galaxy Coevolution, Astrophys. J. (2025) Vol. 979, 212. See 4.1.
P. L. Krapivsky, Hiring Strategies, arXiv:2412.10490 [cs.GT], 2024. See p. 12.
Qipeng Kuang, Ondřej Kuželka, Yuanhong Wang, and Yuyi Wang, Bridging Weighted First Order Model Counting and Graph Polynomials, arXiv:2407.11877 [cs.LO], 2024. See p. 33.
Jack Kuipers and Giusi Moffa, Uniform generation of random acyclic digraphs, arXiv preprint arXiv:1202.6590 [stat.CO], 2012. - _N. J. A. Sloane_, Sep 14 2012
Laphou Lao, Zecheng Li, Songlin Hou, Bin Xiao, Songtao Guo, and Yuanyuan Yang, A Survey of IoT Applications in Blockchain Systems: Architecture, Consensus and Traffic Modeling, ACM Computing Surveys (CSUR, 2020) Vol. 53, No. 1, Article No. 18.
Tobias Ellegaard Larsen, Claus Thorn Ekstrøm, and Anne Helby Petersen, Score-Based Causal Discovery with Temporal Background Information, arXiv:2502.06232 [stat.ME], 2025. See p. 9.
Daniel Mallia, Towards an Unsupervised Bayesian Network Pipeline for Explainable Prediction, Decision Making and Discovery, Master's Thesis, CUNY Hunter College (2023).
B. D. McKay, F. E. Oggier, G. F. Royle, N. J. A. Sloane, I. M. Wanless and H. S. Wilf, Acyclic digraphs and eigenvalues of (0,1)-matrices, J. Integer Sequences, 7 (2004), #04.3.3.
B. D. McKay, F. E. Oggier, G. F. Royle, N. J. A. Sloane, I. M. Wanless and H. S. Wilf, Acyclic digraphs and eigenvalues of (0,1)-matrices, arXiv:math/0310423 [math.CO], Oct 28 2003.
A. Motzek and R. Möller, Exploiting Innocuousness in Bayesian Networks, Preprint 2015.
Yisu Peng, Y. Jiang, and P. Radivojac, Enumerating consistent subgraphs of directed acyclic graphs: an insight into biomedical ontologies, arXiv preprint arXiv:1712.09679 [cs.DS], 2017.
J. Peters, J. Mooij, D. Janzing, and B. Schölkopf, Causal Discovery with Continuous Additive Noise Models, arXiv preprint arXiv:1309.6779 [stat.ML], 2013.
R. W. Robinson, Enumeration of acyclic digraphs, Manuscript. (Annotated scanned copy)
V. I. Rodionov, On the number of labeled acyclic digraphs, Disc. Math. 105 (1-3) (1992) 319-321
I. Shpitser, T. S. Richardson, J. M. Robins and R. Evans, Parameter and Structure Learning in Nested Markov Models, arXiv preprint arXiv:1207.5058 [stat.ML], 2012.
I. Shpitser, R. J. Evans, T. S. Richardson, and J. M. Robins, Introduction to nested Markov models, Behaviormetrika, Behaviormetrika Vol. 41, No. 1, 2014, 3-39.
R. P. Stanley, Acyclic orientation of graphs, Discrete Math. 5 (1973), 171-178. North Holland Publishing Company.
Christian Toth, Christian Knoll, Franz Pernkopf, and Robert Peharz, Rao-Blackwellising Bayesian Causal Inference, arXiv:2402.14781 [cs.LG], 2024.
Sumanth Varambally, Yi-An Ma, and Rose Yu, Discovering Mixtures of Structural Causal Models from Time Series Data, arXiv:2310.06312 [cs.LG], 2023.
S. Wagner, Asymptotic enumeration of extensional acyclic digraphs, in Proceedings of the SIAM Meeting on Analytic Algorithmics and Combinatorics (ANALCO12).
Daniel Waxman, Kurt Butler, and Petar M. Djuric, Dagma-DCE: Interpretable, Non-Parametric Differentiable Causal Discovery, arXiv:2401.02930 [cs.LG], 2024.
Eric Weisstein's World of Mathematics, (0,1)-Matrix
Eric Weisstein's World of Mathematics, Acyclic Digraph
Eric Weisstein's World of Mathematics, Positive Eigenvalued Matrix
Eric Weisstein's World of Mathematics, Weisstein's Conjecture
Jun Wu and Mathias Drton, Partial Homoscedasticity in Causal Discovery with Linear Models, arXiv:2308.08959 [math.ST], 2023.
Chris Ying, Enumerating Unique Computational Graphs via an Iterative Graph Invariant, arXiv:1902.06192 [cs.DM], 2019.
Index entries for sequences related to binary matrices

Cf. A086510, A081064 (refined by # arcs), A307049 (by # descents).
Cf. A055165, which counts nonsingular {0, 1} matrices and A085656, which counts positive definite {0, 1} matrices.
Cf. A188457, A135079, A137435 (acyclic 3-multidigraphs), A188490.
For a unique sink we have A003025.
The unlabeled version is A003087.
These are the reverse-alternating sums of rows of A046860.
The weakly connected case is A082402.
A reciprocal version is A334282.
Row sums of A361718.
Cf. A003465, A058877, A058891, A059201, A088957, A133686, A323818.

p:=evalf(solve(sum((-1)^n*x^n/(n!*2^(n*(n-1)/2)), n=0..infinity) = 0, x), 50); M:=evalf(sum((-1)^(n+1)*p^n/((n-1)!*2^(n*(n-1)/2)), n=1..infinity), 40); # program for evaluation of constants p and M in the asymptotic formula, Vaclav Kotesovec, Dec 09 2013

a[0] = a[1] = 1; a[n_] := a[n] = Sum[ -(-1)^k * Binomial[n, k] * 2^(k*(n-k)) * a[n-k], {k, 1, n}]; Table[a[n], {n, 0, 13}](* Jean-François Alcover, May 21 2012, after PARI *)
Table[2^(n*(n-1)/2)*n! * SeriesCoefficient[1/Sum[(-1)^k*x^k/k!/2^(k*(k-1)/2),{k,0,n}],{x,0,n}],{n,0,20}] (* Vaclav Kotesovec, May 19 2015 *)
Table[Length[Select[Subsets[Subsets[Range[n]],{n}],Length[Select[Tuples[#],UnsameQ@@#&]]==1&]],{n,0,5}] (* Gus Wiseman, Jan 01 2024 *)

a(n)=if(n<1,n==0,sum(k=1,n,-(-1)^k*binomial(n,k)*2^(k*(n-k))*a(n-k)))

{a(n)=polcoeff(1-sum(k=0, n-1, a(k)*x^k/(1+2^k*x+x*O(x^n))^(k+1)), n)} \\ Paul D. Hanna, Oct 17 2009

a(0) = 1; for n > 0, a(n) = Sum_{k=1..n} (-1)^(k+1)*C(n, k)*2^(k*(n-k))*a(n-k).
1 = Sum_{n>=0} a(n)*exp(-2^n*x)*x^n/n!. - Vladeta Jovovic, Jun 05 2005
a(n) = Sum_{k=1..n} (-1)^(n-k)*A046860(n,k) = Sum_{k=1..n} (-1)^(n-k)*k!*A058843(n,k). - Vladeta Jovovic, Jun 20 2008
1 = Sum_{n=>0} a(n)*x^n/(1 + 2^n*x)^(n+1). - Paul D. Hanna, Oct 17 2009
1 = Sum_{n>=0} a(n)*C(n+m-1,n)*x^n/(1 + 2^n*x)^(n+m) for m>=1. - Paul D. Hanna, Apr 01 2011
log(1+x) = Sum_{n>=1} a(n)*(x^n/n)/(1 + 2^n*x)^n. - Paul D. Hanna, Apr 01 2011
Let E(x) = Sum_{n >= 0} x^n/(n!*2^C(n,2)). Then a generating function for this sequence is 1/E(-x) = Sum_{n >= 0} a(n)*x^n/(n!*2^C(n,2)) = 1 + x + 3*x^2/(2!*2) + 25*x^3/(3!*2^3) + 543*x^4/(4!*2^6) + ... (Stanley). Cf. A188457. - Peter Bala, Apr 01 2013
a(n) ~ n!*2^(n*(n-1)/2)/(M*p^n), where p = 1.488078545599710294656246... is the root of the equation Sum_{n>=0} (-1)^n*p^n/(n!*2^(n*(n-1)/2)) = 0, and M = Sum_{n>=1} (-1)^(n+1)*p^n/((n-1)!*2^(n*(n-1)/2)) = 0.57436237330931147691667... Both references to the article "Acyclic digraphs and eigenvalues of (0,1)-matrices" give the wrong value M=0.474! - Vaclav Kotesovec, Dec 09 2013 [Response from N. J. A. Sloane, Dec 11 2013: The value 0.474 has a typo, it should have been 0.574. The value was taken from Stanley's 1973 paper.]
exp( Sum_{n >= 1} a(n)*x^n/n ) = 1 + x + 2*x^2 + 10*x^3 + 146*x^4 + 6010*x^5 + ... appears to have integer coefficients (cf. A188490). - Peter Bala, Jan 14 2016

A001586 Generalized Euler numbers, or Springer numbers.

Original entry on oeis.org

1, 1, 3, 11, 57, 361, 2763, 24611, 250737, 2873041, 36581523, 512343611, 7828053417, 129570724921, 2309644635483, 44110959165011, 898621108880097, 19450718635716001, 445777636063460643, 10784052561125704811, 274613643571568682777, 7342627959965776406281

Offset: 0

N. J. A. Sloane

From Peter Bala, Feb 02 2011: (Start)
The Springer numbers were originally considered by Glaisher (see references). They are a type B analog of the zigzag numbers A000111 for the group of signed permutations.
COMBINATORIAL INTERPRETATIONS
Several combinatorial interpretations of the Springer numbers are known:
1) a(n) gives the number of Weyl chambers in the principal Springer cone of the Coxeter group B_n of symmetries of an n dimensional cube. An example can be found in [Arnold - The Calculus of snakes...].
2) Arnold found an alternative combinatorial interpretation of the Springer numbers in terms of snakes. Snakes are a generalization of alternating permutations to the group of signed permutations. A signed permutation is a sequence (x_1,x_2,...,x_n) of integers such that {|x_1|,|x_2|,...,|x_n|} = {1,2,...,n}. They form a group, the hyperoctahedral group of order 2^n*n! = A000165(n), isomorphic to the group of symmetries of the n dimensional cube. A snake of type B_n is a signed permutation (x_1,x_2,...,x_n) such that 0 < x_1  > x_2 < ... x_n. For example, (3,-4,-2,-5,1,-6) is a snake of type B_6. a(n) gives the number of snakes of type B_n [Arnold]. The cases n=2 and n=3 are given in the Example section below.
3) The Springer numbers also arise in the study of the critical points of functions; they count the topological types of odd functions with 2*n critical values [Arnold, Theorem 35].
4) Let F_n be the set of plane rooted forests satisfying the following conditions:
... each root has exactly one child, and each of the other internal nodes has exactly two (ordered) children,
... there are n nodes labeled by integers from 1 to n, but some leaves can be non-labeled (these are called empty leaves), and labels are increasing from each root down to the leaves. Then a(n) equals the cardinality of F_n. An example and proof are given in [Verges, Theorem 4.5].
OTHER APPEARANCES OF THE SPRINGER NUMBERS
1) Hoffman has given a connection between Springer numbers, snakes and the successive derivatives of the secant and tangent functions.
2) For integer N the quarter Gauss sums Q(N) are defined by ... Q(N) := Sum_{r = 0..floor(N/4)} exp(2*Pi*I*r^2/N). In the cases N = 1 (mod 4) and N = 3 (mod 4) an asymptotic series for Q(N) as N -> inf that involves the Springer numbers has been given by Evans et al., see 1.32 and 1.33.
For a sequence of polynomials related to the Springer numbers see A185417. For a table to recursively compute the Springer numbers see A185418.
(End)
Similar to the way in which the signed Euler numbers A122045 are 2^n times the value of the Euler polynomials at 1/2, the generalized signed Euler numbers A188458 can be seen as 2^n times the value of generalized Euler polynomials at 1/2. These are the Swiss-Knife polynomials A153641. A recursive definition of these polynomials is given in A081658. - Peter Luschny, Jul 19 2012
a(n) is the number of reverse-complementary updown permutations of [2n]. For example, the updown permutation 241635 is reverse-complementary because its complement is 536142, which is the same as its reverse, and a(2)=3 counts 1324, 2413, 3412. - David Callan, Nov 29 2012
a(n) = |2^n G(n,1/2;-1)|, a specialization of the Appell sequence of polynomials umbrally formed by G(n,x;t) = (G(.,0;t) + x)^n from the Grassmann polynomials G(n,0;t) of A046802 enumerating the cells of the positive Grassmannians. - Tom Copeland, Oct 14 2015
Named "Springer numbers" after the Dutch mathematician Tonny Albert Springer (1926-2011). - Amiram Eldar, Jun 13 2021

			a(2) = 3: The three snakes of type B_2 are
  (1,-2), (2,1), (2,-1).
a(3) = 11: The 11 snakes of type B_3 are
  (1,-2,3), (1,-3,2), (1,-3,-2),
  (2,1,3), (2,-1,3), (2,-3,1), (2,-3,-1),
  (3,1,2), (3,-1,2), (3,-2,1), (3,-2,-1).

V. I. Arnold, Springer numbers and Morsification spaces. J. Algebraic Geom., Vol. 1, No. 2 (1992), pp. 197-214.
J. W. L. Glaisher, "On the coefficients in the expansions of cos x/cos 2x and sin x/cos 2x", Quart. J. Pure and Applied Math., Vol. 45 (1914), pp. 187-222.
J. W. L. Glaisher, On the Bernoullian function, Q. J. Pure Appl. Math., Vol. 29 (1898), pp. 1-168.
Ulrike Sattler, Decidable classes of formal power series with nice closure properties, Diplomarbeit im Fach Informatik, Univ. Erlangen - Nürnberg, Jul 27 1994.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
Tonny Albert Springer, Remarks on a combinatorial problem, Nieuw Arch. Wisk., Vol. 19, No. 3 (1971), pp. 30-36.

Matthew House, Table of n, a(n) for n = 0..432 (terms 0..100 from T. D. Noe)
Vladimir Igorevich Arnol'd, The calculus of snakes and the combinatorics of Bernoulli, Euler and Springer numbers of Coxeter groups, Uspekhi Mat. nauk., Vol. 47, No. 1 (1992), pp. 3-45; English version, Russian Math. Surveys, Vol. 47 (1992), pp. 1-51.
Marilena Barnabei, Flavio Bonetti, Niccolò Castronuovo and Matteo Silimbani, Ascending runs in permutations and valued Dyck paths, Ars Mathematica Contemporanea, Vol. 16, No. 2 (2019), pp. 445-463.
Paul Barry, A Note on Three Families of Orthogonal Polynomials defined by Circular Functions, and Their Moment Sequences, Journal of Integer Sequences, Vol. 15 (2012), Article 12.7.2.
Paul Barry, On a transformation of Riordan moment sequences, arXiv:1802.03443 [math.CO], 2018.
William Y. C. Chen, Neil J. Y. Fan and Jeffrey Y. T. Jia, Labeled Ballot Paths and the Springer Numbers, arXiv:1009.2233 [math.CO], Sep 12 2010. [From _Jonathan Vos Post_, Sep 13 2010]
Shaoshi Chen, Hanqian Fang, Sergey Kitaev, and Candice X.T. Zhang, Patterns in Multi-dimensional Permutations, arXiv:2411.02897 [math.CO], 2024. See p. 12.
Suyoung Choi, Boram Park and Hanchul Park, The Betti numbers of real toric varieties associated to Weyl chambers of type B, arXiv preprint arXiv:1602.05406 [math.AT], 2016.
Chak-On Chow and Shi-Mei Ma, Counting signed permutations by their alternating runs, Discrete Mathematics, Vol. 323 (28 May 2014), pp. 49-57.
D. Dumont, Further triangles of Seidel-Arnold type and continued fractions related to Euler and Springer numbers Adv. Appl. Math., Vol. 16, No. 3 (1995), pp. 275-296.
Ronald Evans, Marvin Minei and Bennet Yee, Incomplete higher order Gauss sums, J. Math. Anal. Appl., Vol. 281, No. 2 (2003), pp. 454-476. See 1.32 and 1.33.
Dominique Foata and Guo-Niu Han, Multivariable Tangent and Secant q-derivative Polynomials, arXiv:1304.2486 [math.CO], 2013; alternative link.
J. W. L. Glaisher, On a set of coefficients analogous to the Eulerian numbers, Proc. London Math. Soc., 31 (1899), 216-235.
Tian Han, Sergey Kitaev, and Philip B. Zhang, Distribution of maxima and minima statistics on alternating permutations, Springer numbers, and avoidance of flat POPs, arXiv:2408.12865 [math.CO], 2024. See p. 2.
Christopher R. H. Hanusa, Alejandro H. Morales, and Martha Yip, Column convex matrices, G-cyclic orders, and flow polytopes, arXiv:2107.07326 [math.CO], 2021.
Michael E. Hoffman, Derivative Polynomials, Euler Polynomials, and Associated Integer Sequences, The Electronic Journal of Combinatorics, Vol. 6 (1999), #R21 (see Th. 3.1).
Matthieu Josuat-Vergès, Enumeration of snakes and cycle-alternating permutations, arXiv:1011.0929 [math.CO], 2010.
Matthieu Josuat-Vergès, J.-C. Novelli and J.-Y. Thibon, The algebraic combinatorics of snakes, arXiv preprint arXiv:1110.5272 [math.CO], 2011.
Emanuele Munarini, Two-Parameter Identities for q-Appell Polynomials, Journal of Integer Sequences, Vol. 26 (2023), Article 23.3.1.
Igor Pak and Andrew Soffer, On Higman's k(U_n(F_q)) conjecture, arXiv preprint arXiv:1507.00411 [math.CO], 2015.
Daniel Shanks, Generalized Euler and class numbers, Math. Comp., Vol. 21, No. 100 (1967), pp. 689-694; Vol. 22, No. 103 (1968), p. 699. [Annotated scanned copy]
Daniel Shanks, Generalized Euler and class numbers. Math. Comp., Vol. 21, No. 100 (1967), pp. 689-694.
Daniel Shanks, Corrigenda to: "Generalized Euler and class numbers", Math. Comp. 22, No. 103 (1968), p. 699.
Alan D. Sokal, The Euler and Springer numbers as moment sequences, arXiv:1804.04498 [math.CO], 2018.
Zhi-Hong Sun, Congruences involving Bernoulli polynomials, Discr. Math., Vol. 308, No. 1 (2007), pp. 71-112.
Zhi-Wei Sun, Conjectures involving arithmetical sequences, Number Theory: Arithmetic in Shangri-La (eds., S. Kanemitsu, H.-Z. Li and J.-Y. Liu), Proc. the 6th China-Japan Sem. Number Theory (Shanghai, August 15-17, 2011), World Sci., Singapore, 2013, pp. 244-258; see Conjectures involving combinatorial sequences, arXiv preprint arXiv:1208.2683 [math.CO], 2012.
M. S. Tokmachev, Correlations Between Elements and Sequences in a Numerical Prism, Bulletin of the South Ural State University, Ser. Mathematics. Mechanics. Physics, Vol. 11, No. 1 (2019), pp. 24-33.
Andrei Vieru, Agoh's conjecture: its proof, its generalizations, its analogues, arXiv preprint arXiv:1107.2938 [math.NT], 2011.
Younghan Yoon, Cohomology of type B real permutohedral varieties, arXiv:2501.09496 [math.AT], 2025. See p. 2.

Cf. A007836, A079858, A185417, A185418, A212435.
Row 2 of A349271.
Bisections are A000281 and A000464. Overview in A349264.
Related polynomials are given in A098432, A081658 and A153641.
Cf. A046802.

a := proc(n) local k; (-1)^iquo(n,2)*add(2^k*binomial(n,k)*euler(k),k=0..n) end; # Peter Luschny, Jul 08 2009
a := n -> (-1)^(n+iquo(n,2))*2^(3*n+1)*(Zeta(0,-n,1/8) - Zeta(0,-n,5/8)):
seq(a(n),n=0..21); # Peter Luschny, Mar 11 2015

n=21; CoefficientList[Series[1/(Cos[x]-Sin[x]), {x, 0, n}], x] * Table[k!, {k, 0, n}] (* Jean-François Alcover, May 18 2011 *)
Table[Abs[Numerator[EulerE[n,1/4]]],{n,0,35}] (* Harvey P. Dale, May 18 2011 *)

{a(n) = if(n<0, 0, n! * polcoeff( 1 / (cos(x + x * O(x^n)) - sin(x + x * O(x^n))), n))}; /* Michael Somos, Feb 03 2004 */

{a(n) = my(an); if(n<2, n>=0, an = vector(n+1, m, 1); for(m=2, n, an[m+1] = 2*an[m] + an[m-1] + sum(k=0, m-3, binomial(m-2, k) * (an[k+1] * an[m-1-k] + 2*an[k+2] * an[m-k] - an[k+3] * an[m-1-k]))); an[n+1])}; /* Michael Somos, Feb 03 2004 */

/* Explicit formula by Peter Bala: */
{a(n)=((1+I)/2)^n*sum(k=0,n,((1-I)/(1+I))^k*sum(j=0,k,(-1)^(k-j)*binomial(n+1,k-j)*(2*j+1)^n))}

@CachedFunction
def p(n,x) :
    if n == 0 : return 1
    w = -1 if n%2 == 0 else 0
    v =  1 if n%2 == 0 else -1
    return v*add(p(k,0)*binomial(n,k)*(x^(n-k)+w) for k in range(n)[::2])
def A001586(n) : return abs(2^n*p(n, 1/2))
[A001586(n) for n in (0..21)] # Peter Luschny, Jul 19 2012

E.g.f.: 1/(cos(x) - sin(x)).
Values at 1 of polynomials Q_n() defined in A104035. - N. J. A. Sloane, Nov 06 2009
a(n) = numerator of abs(Euler(n,1/4)). - N. J. A. Sloane, Nov 07 2009
Let B_n(x) = Sum_{k=0.. n*(n-1)/2} b(n,k)*x^k, where b(n,k) is number of n-node acyclic digraphs with k arcs, cf. A081064; then a(n) = |B_n(-2)|. - Vladeta Jovovic, Jan 25 2005
G.f. A(x) = y satisfies y'^2 = 2y^4 - y^2, y''y = y^2 + 2y'^2. - Michael Somos, Feb 03 2004
a(n) = (-1)^floor(n/2) Sum_{k=0..n} 2^k C(n,k) Euler(k). - Peter Luschny, Jul 08 2009
From Peter Bala, Feb 02 2011: (Start)
(1)... a(n) = ((1 + i)/2)^n*B(n,(1 - i)/(1 + i)), where i = sqrt(-1) and {B(n,x)}n>=0 = [1, 1 + x, 1 + 6*x + x^2, 1 + 23*x + 23*x^2 + x^3, ...] is the sequence of type B Eulerian polynomials - see A060187.
This yields the explicit formula
(2)... a(n) = ((1 + i)/2)^n*Sum_{k = 0..n} ((1 - i)/(1 + i))^k * Sum_{j = 0..k} (-1)^(k-j)*binomial(n+1,k-j)*(2*j + 1)^n.
The result (2) can be used to find congruences satisfied by the Springer numbers. For example, for odd prime p
(3)
... a(p) = 1 (mod p) when p = 4*n + 1
... a(p) = -1 (mod p) when p = 4*n + 3.
(End)
E.g.f.: 1/Q(0) where Q(k) = 1 - x/((2k+1)-x*(2k+1)/(x+(2k+2)/Q(k+1))); (continued fraction). - Sergei N. Gladkovskii, Nov 19 2011
E.g.f.: 2/U(0) where U(k) = 1 + 1/(1 + x/(2*k + 1 -x - (2*k+1)/(2 - x/(x+ (2*k+2)/U(k+1))))); (continued fraction, 5-step). - Sergei N. Gladkovskii, Sep 24 2012
E.g.f.: 1/G(0) where G(k) = 1 - x/(4*k+1 - x*(4*k+1)/(4*k+2 + x + x*(4*k+2)/(4*k+3 - x - x*(4*k+3)/(x + (4*k+4)/G(k+1) )))); (continued fraction, 3rd kind, 5-step). - Sergei N. Gladkovskii, Oct 02 2012
G.f.: 1/G(0) where G(k) = 1 - x*(2*k+1) - 2*x^2*(k+1)*(k+1)/G(k+1); (continued fraction). - Sergei N. Gladkovskii, Jan 11 2013
a(n) = | 2*4^n*lerchphi(-1, -n, 1/4) |. - Peter Luschny, Apr 27 2013
a(n) ~ 4 * n^(n+1/2) * (4/Pi)^n / (sqrt(Pi)*exp(n)). - Vaclav Kotesovec, Oct 07 2013
G.f.: T(0)/(1-x), where T(k) = 1 - 2*x^2*(k+1)^2/( 2*x^2*(k+1)^2 - (1-x-2*x*k)*(1-3*x-2*x*k)/T(k+1) ); (continued fraction). - Sergei N. Gladkovskii, Oct 15 2013
a(n) = (-1)^C(n+1,2)*2^(3*n+1)*(Zeta(-n,1/8)-Zeta(-n,5/8)), where Zeta(a,z) is the generalized Riemann zeta function. - Peter Luschny, Mar 11 2015
E.g.f. A(x) satisfies: A(x) = exp( Integral A(x)/A(-x) dx ). - Paul D. Hanna, Feb 04 2017
E.g.f. A(x) satisfies: A'(x) = A(x)^2/A(-x). - Paul D. Hanna, Feb 04 2017

More terms from Vladeta Jovovic, Jan 25 2005

A055533 Number of labeled order relations on n nodes in which longest chain has n-1 nodes.

Original entry on oeis.org

1, 12, 108, 960, 9000, 90720, 987840, 11612160, 146966400, 1995840000, 28979596800, 448345497600, 7366565606400, 128152088064000, 2353813862400000, 45527990796288000, 925143000477696000

Offset: 2

N. J. A. Sloane, Jul 10 2000

nonn

Also number of labeled acyclic digraphs with n nodes and binomial(n,2)-1 arcs, cf. A081064. - Vladeta Jovovic, Jul 14 2008

L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 60.

A column or diagonal of triangle in A342587.

a(n)=if(n<2,0,n!/2*(n-1)^2) /* Michael Somos, Mar 06 2004 */

E.g.f.: (x^2/2)(1+x)/(1-x)^3. a(n)=n!(n-1)^2/2, if n>1. - Vladeta Jovovic, Dec 01 2002

D-finite with recurrence (n-2)^2*a(n) -n*(n-1)^2*a(n-1)=0. - R. J. Mathar, Mar 08 2021

More terms from Vladeta Jovovic, Dec 01 2002

A147796 Number of consistent sets of 3 irreflexive binary order relationships over n objects.

Original entry on oeis.org

6, 152, 940, 3600, 10570, 26096, 56952, 113280, 209550, 365640, 608036, 971152, 1498770, 2245600, 3278960, 4680576, 6548502, 8999160, 12169500, 16219280, 21333466, 27724752, 35636200, 45344000, 57160350, 71436456, 88565652, 108986640, 133186850, 161705920

Offset: 3

R. H. Hardin, May 04 2009

nonn

From Petros Hadjicostas, Apr 10 2020: (Start)
Here is a proof of the formula for a(n). There are n*(n-1) irreflexive binary order relationships among n distinct objects and binomial(n*(n-1), 3) 3-sets of such relationships.
We first exclude the 3-sets that contain both relationships of the form x < y and y < x (with x <> y), and there are (n*(n-1)/2) *(n*(n-1) - 2) such 3-sets.
Next we exclude the 3-sets that contain all the relationships of the form x < y, y < z, and z < x (with x, y, z distinct), and there are 2*binomial(n,3) of these.
The two groups of excluded 3-sets do not overlap, and the formula has been proved.
It seems that a(n) = A081064(n,3) = number of labeled acyclic directed graphs with n nodes and k = 3 arcs (see Rodionov (1992)). The reason is that we may label the graphs with the n objects and draw an arc from X towards Y if and only if X < Y. The 3 arcs of the directed graph correspond to the 3-set of binary order relationships and they are consistent because the directed graph is acyclic.
(End)

			From _Petros Hadjicostas_, Apr 10 2020: (Start)
For n = 3, consider objects a, b, c. There are 3*2 = 6 irreflexive binary order relationships among these objects (a < b, b < a, a < c, c < a, b < c, c < b). For a set of 3 such sets of binary relationships to be consistent, x < y and y < x should not appear together, and x < y, y < z, and z < x should not be together. We have the following sets of 3 such relationships that are consistent: {x < y, y < z, x < z}, where (x,y,z) is in S_3. Thus, a(3) = |S_3| = 3! = 6. (End)

V. I. Rodionov, On the number of labeled acyclic digraphs, Discr. Math. 105 (1-3) (1992), 319-321.

Related sequences for the number of consistent sets of k irreflexive binary order relationships over n objects: A147817 (k = 4), A147821 (k = 5), A147860 (k = 6), A147872 (k = 7), A147881 (k = 8), A147883 (k = 9), A147964 (k = 10).

Column k = 3 of A081064.

a := n -> (1/6)*n*(n-1)*(n-2)*(n^3-5*n-6):
seq(a(n), n=3..32); # Peter Luschny, Apr 11 2020

a(n) = binomial(n*(n-1), 3) - n*(n-1)*(n*(n-1) - 2)/2 - 2*binomial(n,3) = binomial(n,3) * (n^3 - 5*n - 6). - Petros Hadjicostas, Apr 10 2020
Conjectures from Colin Barker, Apr 11 2020: (Start)
G.f.: 2*x^3*(3 + 55*x + x^2 + x^3) / (1 - x)^7.
a(n) = 7*a(n-1) - 21*a(n-2) + 35*a(n-3) - 35*a(n-4) + 21*a(n-5) - 7*a(n-6) + a(n-7) for n>8.
(End)

More terms from Vaclav Kotesovec, Apr 11 2020

Offset changed to n=3 by Petros Hadjicostas, Apr 11 2020

It seems that a(n) = A081064(n,4) = number of labeled acyclic directed graphs with n nodes and k = 4 arcs (see Rodionov (1992)). The reason is that we may label the graphs with the n objects and draw an arc from X towards Y if and only if X < Y. The 4 arcs of the directed graph correspond to the 4-set of binary order relationships and they are consistent because the directed graph is acyclic. - Petros Hadjicostas, Apr 10 2020

It seems that a(n) = A081064(n,5) is the number of labeled acyclic directed graphs with n nodes and k = 5 arcs (see Rodionov (1992)). The reason is that we may label the graphs with the n objects and draw an arc from X towards Y if and only if X < Y. The 5 arcs of the directed graph correspond to the 3-set of binary order relationships and they are consistent because the directed graph is acyclic. - Petros Hadjicostas, Apr 10 2020

cp's OEIS Frontend

A003024 Number of acyclic digraphs (or DAGs) with n labeled nodes.

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A001586 Generalized Euler numbers, or Springer numbers.

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A055533 Number of labeled order relations on n nodes in which longest chain has n-1 nodes.

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A147796 Number of consistent sets of 3 irreflexive binary order relationships over n objects.

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A147817 Number of consistent sets of 4 irreflexive binary order relationships over n objects.

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A147821 Number of consistent sets of 5 irreflexive binary order relationships over n objects.

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