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%I M1888 #242 Dec 12 2023 17:33:01
%S 1,1,2,8,56,608,9440,198272,5410688,186043904,7867739648,401293838336,
%T 24290513745920,1721379917619200,141174819474169856,
%U 13266093250285568000,1415974941618255921152,170361620874699124637696,22948071824232932086513664,3439933090471867097102680064
%N Genocchi medians (or Genocchi numbers of second kind).
%C 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 The earliest known reference for these numbers is Seidel (1877, pages 185 and 186). - _Don Knuth_, Jul 13 2007
%C Hankel transform of 1,1,2,8,... is A168488. - _Paul Barry_, Nov 27 2009
%C 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 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 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 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H 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 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 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 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 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 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 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 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 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 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 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 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 D. Dumont and J. Zeng, <a href="http://math.univ-lyon1.fr/homes-www/zeng/public_html/paper/publication.html">Polynomes d'Euler et les fractions continues de Stieltjes-Rogers</a>, Ramanujan J. 2 (1998) 3, 387-410.
%H 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 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 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 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 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 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 Gábor Hetyei, <a href="https://arxiv.org/abs/1704.07245">Alternation acyclic tournaments</a>, arXiv:math/1704.07245 [math.CO], 2017.
%H 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 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 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 A. Randrianarivony and J. Zeng, <a href="http://dx.doi.org/10.1006/aama.1996.0001">Une famille de polynomes qui interpole plusieurs suites classiques de nombres</a>, Adv. Appl. Math. 17 (1996), 1-26. In French.
%H 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 Alan Sokal, <a href="/A005439/a005439.txt">Table of n, a(n) for n = 1..10000 [315 MB file]</a>
%H Zhi-Wei Sun, <a href="http://arxiv.org/abs/2108.07723">Arithmetic properties of some permanents</a>, arXiv:2108.07723 [math.GM], 2021.
%H 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 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 a(n) = A000366(n)*2^(n-1).
%F 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 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 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 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 From _Sergei N. Gladkovskii_, Dec 14 2011, Dec 27 2012, May 29 2013, Oct 09 2013, Oct 24 2013, Oct 27 2013: (Start)
%F Continued fractions:
%F 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 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 G.f.: A(x) = (1/(G(0)-1)/x where G(k) = 1 - x*(k+1)^2/(1 - x*(k+1)^2/G(k+1)).
%F G.f.: 2/G(0), where G(k) = 1 + 1/(1 - 1/(1 - 1/(4*x*(k+1)) + 1/G(k+1))).
%F 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 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 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 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 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 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.
%p 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 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 (PARI) a(n)=2*(-1)^n*sum(k=0,n,binomial(n,k)*(1-2^(n+k+1))* bernfrac(n+k+1))
%o (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 (Sage) # Algorithm of L. Seidel (1877)
%o # n -> [a(1), ..., a(n)] for n >= 1.
%o def A005439_list(n) :
%o D = []; [D.append(0) for i in (0..n+2)]; D[1] = 1
%o R = [] ; b = True
%o for i in(0..2*n-1) :
%o h = i//2 + 1
%o if b :
%o for k in range(h-1,0,-1) : D[k] += D[k+1]
%o else :
%o for k in range(1,h+1,1) : D[k] += D[k-1]
%o if b : R.append(D[1])
%o b = not b
%o return R
%o A005439_list(18) # _Peter Luschny_, Apr 01 2012
%o (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 (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 (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 (Python)
%o from math import comb
%o from sympy import bernoulli
%o 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 Cf. A000366, A036968, A110501, A297703.
%K nonn,nice,easy
%O 0,3
%A _Simon Plouffe_
%E More terms and additional comments from _David W. Wilson_, Jan 11 2001
%E a(0)=1 prepended by _Peter Luschny_, Apr 14 2023
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