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%I #113 Jul 13 2023 07:01:38
%S 1,1,2,7,38,295,3098,42271,726734,15366679,391888514,11860602415,
%T 420258768950,17233254330343,809698074358250,43212125903877439,
%U 2599512037272630686,175079893678534943287,13122303354155987156306
%N Genocchi numbers of second kind (A005439) divided by 2^(n-1).
%C The earliest known reference to these numbers is the Dellac Marseille memoir. - _Don Knuth_, Jul 11 2007
%C According to Ira Gessel, Dellac's interpretation is the following: start with a 2n X n array of cells and consider the set D of cells in rows i through i+n of column i, for i from 1 to n. Then a(n) is the number of subsets of D containing two cells in each column and one cell in each row.
%C Barsky proved that for even n>1, a(n) is congruent to 3 mod 4 and for odd n>1, congruent to 2 mod 4. Gessel shows that for even n>5, a(n) is congruent to 4n-1 mod 16 and for odd n>2 that a(n)/2 is congruent to 2-n mod 8.
%C The entry for A005439 has further information.
%C The number of sequences (I_1,...,I_{n-1}) consisting of subsets of the set {1,...,n} such that the number of elements in I_k is exactly k and I_k\subset I_{k+1}\cup {k+1}. The Euler characteristics of the degenerate flag varieties of type A. - _Evgeny Feigin_, Dec 15 2011
%C Kreweras proved that for n>2, a(n) is alternatively congruent to 2 and to 7 mod 36. - _Michel Marcus_, Nov 06 2012
%D Anonymous, L'Intermédiaire des Mathématiciens, 7 (1900), p. 328.
%D Hippolyte Dellac, Problem 1735, L'Intermédiaire des Mathématiciens, Vol. 7 (1900), p. 9 ff.
%D E. Lemoine, L'Intermédiaire des Mathématiciens, 8 (1901), 168-169.
%D Qiongqiong Pan and Jiang Zeng, Cycles of even-odd drop permutations and continued fractions of Genocchi numbers, arXiv:2108.03200 [math.CO], 2021.
%D L. Seidel, Über eine einfache Entstehungsweise der Bernoulli'schen Zahlen und einiger verwandten Reihen, Sitzungsberichte der mathematisch-physikalischen Classe der königlich bayerischen Akademie der Wissenschaften zu München, volume 7 (1877), 157-187.
%H T. D. Noe, <a href="/A000366/b000366.txt">Table of n, a(n) for n=1..100</a>
%H D. Barsky, <a href="http://www.numdam.org/item?id=GAU_1979-1981__7-8__A15_0">Congruences pour les nombres de Genocchi de 2e espèce</a>, Groupe d'étude d'Analyse ultramétrique, 8e année, no. 34, 1980/81, 13 pp.
%H Ange Bigeni, <a href="https://arxiv.org/abs/1705.03804">Enumerating the symplectic Dellac configurations</a>, arXiv:1705.03804 [math.CO], 2017.
%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 Ange Bigeni and Evgeny Feigin, <a href="https://arxiv.org/abs/1808.04275">Symmetric Dellac configurations</a>, arXiv:1808.04275 [math.CO], 2018.
%H Hippolyte Dellac, <a href="https://odyssee.univ-amu.fr/files/original/2/18/Annales-faculte-sc-Mrs_1901_T-11.pdf">Note sur l'élimination, méthode de parallélogramme</a>, Annales de la Faculté des Sciences de Marseille, XI (1901), 141-164.
%H E. Feigin, <a href="https://arxiv.org/abs/1101.1898">Degenerate flag varieties and the median Genocchi numbers</a>, arXiv:1101.1898 [math.AG], 2011.
%H E. Feigin, <a href="https://arxiv.org/abs/1111.0740">The median Genocchi numbers, Q-analogues and continued fractions</a>, arXiv:1111.0740 [math.CO], 2011-2012.
%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 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 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.
%H G. Kreweras and D. Dumont, <a href="http://dx.doi.org/10.1016/S0012-365X(99)00238-1">Sur les anagrammes alternés</a>, Discrete Mathematics, Volume 211, Issues 1-3, 28 January 2000, Pages 103-110.
%F From _Don Knuth_, Jul 11 2007: (Start)
%F The anonymous 1900 note in Interm. Math. gives a formula that is equivalent to a nice generating function:
%F For example, the first four terms on the right are
%F 1
%F ... 2x - 2x^2 + 2x^3 + ...
%F ........ 9x^2 - 36x^3 + ...
%F ............... 72x^3 + ...
%F summing to 1 + 2x + 7x^2 + 38x^3 + ... . Of course one can replace x with 2x and get a generating function for A005439. (End)
%F (-2)^(2-n) * Sum_{k=0..n} C(n, k)*(1-2^(n+k+1))*B(n+k+1), with B(n) the Bernoulli numbers.
%F O.g.f.: A(x) = x/(1-x/(1-x/(1-3*x/(1-3*x/(1-6*x/(1-6*x/(... -[n/2+1]*[n/2+2]/2*x/(1- ...)))))))) (continued fraction). - _Paul D. Hanna_, Oct 07 2005
%F Sum_{n>0} a(n)x^n = Sum_{n>0} (n!^2/2^{n-1}) (x^n/((1+x)(1+3x)...(1+binomial(n,2)x))).
%F a(n+1) = Sum_{k=0..n} A211183(n,k). - _Philippe Deléham_, Feb 03 2013
%F G.f.: Q(0)*2 - 2, where Q(k) = 1 - x*(k+1)^2/( x*(k+1)^2 - 2/(1 - x*(k+1)^2/( x*(k+1)^2 - 2/Q(k+1) ))); (continued fraction). - _Sergei N. Gladkovskii_, Oct 22 2013
%F a(n) ~ 2^(n+5) * n^(2*n+3/2) / (exp(2*n) * Pi^(2*n+1/2)). - _Vaclav Kotesovec_, Oct 28 2014
%e G.f. = x + x^2 + 2*x^3 + 7*x^4 + 38*x^5 + 295*x^6 + 3098*x^7 + ...
%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,19}] (* _Jean-François Alcover_, Jul 18 2011, after PARI prog. *)
%o (PARI) a(n)=(-1/2)^(n-2)*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));if(n<1,return(0), for(k=1,n,CF=1/(1-((n-k)\2+1)*((n-k)\2+2)/2*x*CF));return(Vec(CF)[n]))} (Hanna)
%o (PARI) {a(n)=polcoeff( x*sum(m=0, n, m!*(m+1)!*(x/2)^m / prod(k=1, m,1 + k*(k+1)*x/2 +x*O(x^n)) ), n)} \\ _Paul D. Hanna_, Feb 03 2013
%o (Sage) # Algorithm of L. Seidel (1877)
%o # n -> [a(1), ..., a(n)] for n >= 1.
%o def A000366_list(n) :
%o D = [0]*(n+2); D[1] = 1
%o R = []; z = 1/2; b = False
%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 z *= 2
%o else :
%o for k in range(1, h+1, 1) : D[k] += D[k-1]
%o b = not b
%o if not b : R.append(D[1]/z)
%o return R
%o A000366_list(19) # _Peter Luschny_, Jun 29 2012
%o (Python)
%o from math import comb
%o from sympy import bernoulli
%o def A000366(n): return (-1 if n&1 else 1)*sum(comb(n,k)*(1-(1<<n+k+1))*bernoulli(n+k+1) for k in range(n+1))>>n-2 if n>1 else 1 # _Chai Wah Wu_, Apr 14 2023
%Y Cf. A001469, A005439, A130168, A130169.
%Y First column, first diagonal and row sums of triangle A014784.
%Y Also row sums of triangle A239894.
%K nonn,easy,nice
%O 1,3
%A _Don Knuth_, _N. J. A. Sloane_
%E More terms from _David W. Wilson_, Jan 11 2001
%E Edited by _Ralf Stephan_, Apr 17 2004
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