This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A005439 M1888 #267 Aug 26 2025 14:24:39
%S A005439 1,1,2,8,56,608,9440,198272,5410688,186043904,7867739648,401293838336,
%T A005439 24290513745920,1721379917619200,141174819474169856,
%U A005439 13266093250285568000,1415974941618255921152,170361620874699124637696,22948071824232932086513664,3439933090471867097102680064
%N A005439 Genocchi medians (or Genocchi numbers of second kind).
%C A005439 a(n) is the number of Boolean functions of n variables whose ROBDD (reduced ordered binary decision diagram) contains exactly n branch nodes, one for each variable. - _Don Knuth_, Jul 11 2007
%C A005439 The earliest known reference for these numbers is Seidel (1877, pages 185 and 186). - _Don Knuth_, Jul 13 2007
%C A005439 Hankel transform of 1,1,2,8,... is A168488. - _Paul Barry_, Nov 27 2009
%C A005439 According to Hetyei [2017], alternation acyclic tournaments "are counted by the median Genocchi numbers"; an alternation acyclic tournament "does not contain a cycle in which descents and ascents alternate." - _Danny Rorabaugh_, Apr 25 2017
%C A005439 The n-th Genocchi number of the second kind is also the number of collapsed permutations in (2n) letters. A permutation pi of size 2n is said to be collapsed if 1+floor(k/2) <= pi^{-1}(k) <= n + floor(k/2). There are 2 collapsed permutations of size 4, namely 1234 and 1324. - _Arvind Ayyer_, Oct 23 2020
%C A005439 For any positive integer n, a(n) is (-1)^n times the permanent of the 2n X 2n matrix M with M(j, k) = floor((2*j-k-1)/(2*n)). This former conjecture of Luschny, inspired by a conjecture of _Zhi-Wei Sun_ in A036968, was proven by Fu, Lin and Sun (see link). - _Peter Luschny_, Sep 07 2021 [updated Sep 24 2021]
%D A005439 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H A005439 Muniru A Asiru, <a href="/A005439/b005439.txt">Table of n, a(n) for n = 0..270</a> (terms n = 1..100 from T. D. Noe)
%H A005439 A. Ayyer, D. Hathcock, and P. Tetali, <a href="https://arxiv.org/abs/2010.11236">Toppleable Permutations, Excedances and Acyclic Orientations</a>, arXiv:2010.11236 [math.CO], 2020.
%H A005439 Paul Barry, <a href="https://arxiv.org/abs/2107.14278">Series reversion with Jacobi and Thron continued fractions</a>, arXiv:2107.14278 [math.NT], 2021.
%H A005439 Beáta Bényi, <a href="https://doi.org/10.1007/s00373-021-02442-2">A Bijection for the Boolean Numbers of Ferrers Graphs</a>, Graphs and Combinatorics (2022) Vol. 38, No. 10.
%H A005439 Ange Bigeni, <a href="https://arxiv.org/abs/1712.05475">The universal sl2 weight system and the Kreweras triangle</a>, arXiv:1712.05475 [math.CO], 2017.
%H A005439 Ange Bigeni, <a href="https://arxiv.org/abs/1712.01929">Combinatorial interpretations of the Kreweras triangle in terms of subset tuples</a>, arXiv:1712.01929 [math.CO], 2017.
%H A005439 Ange Bigeni, <a href="https://doi.org/10.1016/j.jcta.2018.08.005">A generalization of the Kreweras triangle through the universal sl_2 weight system</a>, Journal of Combinatorial Theory, Series A (2019) Vol. 161, 309-326.
%H A005439 Alin Bostan, Lucia Di Vizio, and Kilian Raschel, <a href="https://arxiv.org/abs/2504.13542">Singular walks in the quarter plane and Bernoulli numbers</a>, arXiv:2504.13542 [math.CO], 2025. See p. 28.
%H A005439 Alexander Burstein, Sergi Elizalde, and Toufik Mansour, <a href="https://arxiv.org/abs/math/0610234">Restricted Dumont permutations, Dyck paths and noncrossing partitions</a>, arXiv:math/0610234 [math.CO], 2006. [Theorem 3.5]
%H A005439 Kwang-Wu Chen, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL6/Chen/chen50.html">An Interesting Lemma for Regular C-fractions</a>, J. Integer Seqs., Vol. 6, 2003.
%H A005439 Shane Chern, <a href="https://arxiv.org/abs/2112.02074">Parity considerations for drops in cycles on {1,2,...,n}</a>, arXiv:2112.02074 [math.CO], 2021.
%H A005439 Bishal Deb and Alan D. Sokal, <a href="https://arxiv.org/abs/2212.07232">Classical continued fractions for some multivariate polynomials generalizing the Genocchi and median Genocchi numbers</a>, arXiv:2212.07232 [math.CO], 2022. See pp. 14-15.
%H A005439 Bishal Deb, <a href="https://arxiv.org/abs/2304.14487">Continued fractions using a Laguerre digraph interpretation of the Foata-Zeilberger bijection and its variants</a>, arXiv:2304.14487 [math.CO], 2023. See p. 4.
%H A005439 Bishal Deb, <a href="https://arxiv.org/abs/2508.13709">Cyclic sieving phenomena via combinatorics of continued fractions</a>, arXiv:2508.13709 [math.CO], 2025. See p. 38.
%H A005439 D. Dumont and J. Zeng, <a href="http://math.univ-lyon1.fr/homes-www/zeng/public_html/paper/publication.html">Polynômes d'Euler et les fractions continues de Stieltjes-Rogers</a>, Ramanujan J. 2 (1998) 3, 387-410.
%H A005439 Richard Ehrenborg and Einar Steingrímsson, <a href="http://dx.doi.org/10.1006/eujc.1999.0370">Yet another triangle for the Genocchi numbers</a>, European J. Combin. 21 (2000), no. 5, 593-600. MR1771988 (2001h:05008).
%H A005439 Sen-Peng Eu, Tung-Shan Fu, Hsin-Hao Lai, and Yuan-Hsun Lo, <a href="https://arxiv.org/abs/2103.09130">Gamma-positivity for a Refinement of Median Genocchi Numbers</a>, arXiv:2103.09130 [math.CO], 2021.
%H A005439 Vincent Froese and Malte Renken, <a href="https://arxiv.org/abs/2210.16281">Terrain-like Graphs and the Median Genocchi Numbers</a>, arXiv:2210.16281 [math.CO], 2022.
%H A005439 Shishuo Fu, Zhicong Lin, and Zhi-Wei Sun, <a href="https://arxiv.org/abs/2109.11506">Proofs of five conjectures relating permanents to combinatorial sequences</a>, arXiv:2109.11506 [math.CO], 2021.
%H A005439 Shishuo Fu, Zhicong Lin, and Zhi-Wei Sun, <a href="https://doi.org/10.1016/j.aam.2024.102789">Permanent identities, combinatorial sequences, and permutation statistics</a>, Advances in Applied Mathematics, Volume 163, Part A, 102789 (2025).
%H A005439 I. M. Gessel, <a href="https://arxiv.org/abs/math/0108121">Applications of the classical umbral calculus</a>, arXiv:math/0108121 [math.CO], 2001.
%H A005439 G. Han and J. Zeng, <a href="http://www.labmath.uqam.ca/~annales/volumes/23-1/PDF/063-072.pdf">On a q-sequence that generalizes the median Genocchi numbers</a>, Annal Sci. Math. Quebec, 23(1999), no. 1, 63-72.
%H A005439 Gábor Hetyei, <a href="https://arxiv.org/abs/1704.07245">Alternation acyclic tournaments</a>, arXiv:math/1704.07245 [math.CO], 2017.
%H A005439 G. Kreweras, <a href="http://dx.doi.org/10.1006/eujc.1995.0081">Sur les permutations comptées par les nombres de Genocchi de 1-ière et 2-ième espèce</a>, Europ. J. Comb., vol. 18, pp. 49-58, 1997. (See also page 76.)
%H A005439 Alexander Lazar and Michelle L. Wachs, <a href="https://arxiv.org/abs/1910.07651">The Homogenized Linial Arrangement and Genocchi Numbers</a>, arXiv:1910.07651 [math.CO], 2019.
%H A005439 Qiongqiong Pan and Jiang Zeng, <a href="https://arxiv.org/abs/2108.03200">Cycles of even-odd drop permutations and continued fractions of Genocchi numbers</a>, arXiv:2108.03200 [math.CO], 2021.
%H A005439 A. Randrianarivony and J. Zeng, <a href="http://dx.doi.org/10.1006/aama.1996.0001">Une famille de polynômes qui interpole plusieurs suites classiques de nombres</a>, Adv. Appl. Math. 17 (1996), 1-26. In French.
%H A005439 L. Seidel, <a href="http://publikationen.badw.de/de/003384831">Über eine einfache Entstehungsweise der Bernoulli'schen Zahlen und einiger verwandten Reihen</a>, Sitzungsberichte der mathematisch-physikalischen Classe der königlich bayerischen Akademie der Wissenschaften zu München, volume 7 (1877), 157-187.
%H A005439 Alan Sokal, <a href="/A005439/a005439.txt">Table of n, a(n) for n = 1..10000 [315 MB file]</a>
%H A005439 Zhi-Wei Sun, <a href="http://arxiv.org/abs/2108.07723">Arithmetic properties of some permanents</a>, arXiv:2108.07723 [math.GM], 2021.
%H A005439 G. Viennot, <a href="http://www.jstor.org/stable/44165433">Interprétations combinatoires des nombres d'Euler et de Genocchi</a>, Seminar on Number Theory, 1981/1982, Exp. No. 11, 94 pp., Univ. Bordeaux I, Talence, 1982.
%F A005439 a(n) = T(n, 1) where T(1, x) = 1; T(n, x) = (x+1)*((x+1)*T(n-1, x+1)-x*T(n-1, x)); see A058942.
%F A005439 a(n) = A000366(n)*2^(n-1).
%F A005439 a(n) = 2 * (-1)^n * Sum_{k=0..n} binomial(n, k)*(1-2^(n+k+1))*B(n+k+1), with B(n) the Bernoulli numbers. - _Ralf Stephan_, Apr 17 2004
%F A005439 O.g.f.: 1 + x*A(x) = 1/(1-x/(1-x/(1-4*x/(1-4*x/(1-9*x/(1-9*x/(... -[(n+1)/2]^2*x/(1-...)))))))) (continued fraction). - _Paul D. Hanna_, Oct 07 2005
%F A005439 G.f.: (of 1,1,2,8,...) 1/(1-x-x^2/(1-5*x-16*x^2/(1-13*x-81*x^2/(1-25*x-256*x^2/(1-41*x-625*x^2/(1-... (continued fraction). - _Paul Barry_, Nov 27 2009
%F A005439 O.g.f.: Sum_{n>=0} n!*(n+1)! * x^(n+1) / Product_{k=1..n} (1 + k*(k+1)*x). - _Paul D. Hanna_, May 10 2012
%F A005439 From _Sergei N. Gladkovskii_, Dec 14 2011, Dec 27 2012, May 29 2013, Oct 09 2013, Oct 24 2013, Oct 27 2013: (Start)
%F A005439 Continued fractions:
%F A005439 G.f.: A(x) = 1/S(0), S(k) = 1 - x*(k+1)*(k+2)/(1 - x*(k+1)*(k+2)/S(k+1)).
%F A005439 G.f.: A(x) = -1/S(0), S(k) = 2*x*(k+1)^2 - 1 - x^2*(k+1)^2*(k+2)^2/S(k+1).
%F A005439 G.f.: A(x) = 1/G(0) where G(k) = 1 - x*(k+1)^2/(1 - x*(k+1)^2/G(k+1)).
%F A005439 G.f.: 2/G(0), where G(k) = 1 + 1/(1 - 1/(1 - 1/(4*x*(k+1)) + 1/G(k+1))).
%F A005439 G.f.: Q(0)/x - 1/x, where Q(k) = 1 - x*(k+1)^2/( x*(k+1)^2 - 1/(1 - x*(k+1)^2/( x*(k+1)^2 - 1/Q(k+1)))).
%F A005439 G.f.: T(0)/(1-2*x), where T(k) = 1 - x^2*((k + 2)*(k+1))^2/(x^2*((k + 2)*(k+1))^2 - (1 - 2*x*k^2 - 4*x*k - 2*x)*(1 - 2*x*k^2 - 8*x*k - 8*x)/T(k+1)).
%F A005439 G.f.: R(0), where R(k) = 1 - x*(k+1)*(k+2)/( x*(k+1)*(k+2) - 1/(1 - x*(k+1)*(k+2)/( x*(k+1)*(k+2) - 1/R(k+1) ))). (End)
%F A005439 a(n) ~ 2^(2*n+4) * n^(2*n+3/2) / (exp(2*n) * Pi^(2*n+1/2)). - _Vaclav Kotesovec_, Oct 28 2014
%F A005439 Rewriting the above: a(n) ~ 4*(2*n+1)! / Pi^(2*n+1). Compare to Genocchi numbers A110501(n) = g_n ~ 4*(2*n)! / Pi^(2*n). So these are indeed like "Genocchi medians" g_{n + 1/2}. - _Alan Sokal_, May 13 2022
%F A005439 Asymptotic expansion: a(n) ~ 4*(2*n+1)! * Pi^(-(2*n+1)) * (1 + (Pi^2/16)/n + (Pi^2 (Pi^2 - 16)/512)/n^2 + (Pi^2 (Pi^4 + 384)/24576)/n^3 + (Pi^2 (Pi^6 + 96*Pi^4 + 768*Pi^2 - 12288)/1572864)/n^4 + (Pi^2 (Pi^8 + 320*Pi^6 + 12800*Pi^4 + 491520)/125829120)/n^5 + ...) --- Proof uses binomial sum for Genocchi medians in terms of Genocchi or Bernoulli numbers, combined with leading term of convergent sum (with exponentially small corrections) for the latter. Can also check against the 10000 term a-file. - _Alan Sokal_, May 23 2022.
%F A005439 a(n) = n!^2 * [x^n*y^n] exp(x)*f(x-y), where f(x) is the derivative of the Genocchi number generating function 2*x/(exp(x)+1). - _Ira M. Gessel_, Jul 23 2024
%p A005439 seq(2*(-1)^n*add(binomial(n,k)*(1 - 2^(n+k+1))*bernoulli(n+k+1), k=0..n), n=0..20); # _G. C. Greubel_, Oct 18 2019
%t A005439 a[n_]:= 2*(-1)^(n-2)*Sum[Binomial[n, k]*(1 -2^(n+k+1))*BernoulliB[n+k+1], {k, 0, n}]; Table[a[n], {n,16}] (* _Jean-François Alcover_, Jul 18 2011, after PARI prog. *)
%o A005439 (PARI) a(n)=2*(-1)^n*sum(k=0,n,binomial(n,k)*(1-2^(n+k+1))* bernfrac(n+k+1))
%o A005439 (PARI) a(n)=local(CF=1+x*O(x^(n+2)));if(n<0,return(0), for(k=1,n+1,CF=1/(1-((n-k+1)\2+1)^2*x*CF));return(Vec(CF)[n+2])) \\ _Paul D. Hanna_
%o A005439 (Sage) # Algorithm of L. Seidel (1877)
%o A005439 # n -> [a(1), ..., a(n)] for n >= 1.
%o A005439 def A005439_list(n) :
%o A005439 D = []; [D.append(0) for i in (0..n+2)]; D[1] = 1
%o A005439 R = [] ; b = True
%o A005439 for i in(0..2*n-1) :
%o A005439 h = i//2 + 1
%o A005439 if b :
%o A005439 for k in range(h-1,0,-1) : D[k] += D[k+1]
%o A005439 else :
%o A005439 for k in range(1,h+1,1) : D[k] += D[k-1]
%o A005439 if b : R.append(D[1])
%o A005439 b = not b
%o A005439 return R
%o A005439 A005439_list(18) # _Peter Luschny_, Apr 01 2012
%o A005439 (Sage) [2*(-1)^n*sum(binomial(n,k)*(1-2^(n+k+1))*bernoulli(n+k+1) for k in (0..n)) for n in (1..20)] # _G. C. Greubel_, Oct 18 2019
%o A005439 (Magma) [2*(-1)^n*(&+[Binomial(n, k)*(1-2^(n+k+1))*Bernoulli(n+k+1): k in [0..n]]): n in [1..20]]; // _G. C. Greubel_, Nov 28 2018
%o A005439 (GAP) List([1..20],n->2*(-1)^n*Sum([0..n],k->Binomial(n,k)*(1-2^(n+k+1))*Bernoulli(n+k+1))); # _Muniru A Asiru_, Nov 29 2018
%o A005439 (Python)
%o A005439 from math import comb
%o A005439 from sympy import bernoulli
%o A005439 def A005439(n): return (-2 if n&1 else 2)*sum(comb(n,k)*(1-(1<<n+k+1))*bernoulli(n+k+1) for k in range(n+1)) # _Chai Wah Wu_, Apr 14 2023
%Y A005439 Cf. A000366, A036968, A110501, A297703.
%K A005439 nonn,nice,easy,changed
%O A005439 0,3
%A A005439 _Simon Plouffe_
%E A005439 More terms and additional comments from _David W. Wilson_, Jan 11 2001
%E A005439 a(0)=1 prepended by _Peter Luschny_, Apr 14 2023