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A001469 Genocchi numbers (of first kind); unsigned coefficients give expansion of x*tan(x/2).
(Formerly M3041 N1233)
73

%I M3041 N1233

%S -1,1,-3,17,-155,2073,-38227,929569,-28820619,1109652905,-51943281731,

%T 2905151042481,-191329672483963,14655626154768697,

%U -1291885088448017715,129848163681107301953,-14761446733784164001387

%N Genocchi numbers (of first kind); unsigned coefficients give expansion of x*tan(x/2).

%C The Genocchi numbers satisfy Seidel's recurrence: for n>1, 0 = Sum_{j=0..[n/2]} C(n,2j)*a(n-j). - _Ralf Stephan_, Apr 17 2004

%C The (n+1)st Genocchi number is the number of Dumont permutations of the first kind on 2n letters. In a Dumont permutation of the first kind, each even integer must be followed by a smaller integer and each odd integer is either followed by a larger integer or is the last element. - _Ralf Stephan_, Apr 26 2004

%C According to Hetyei [2017], "alternation acyclic tournaments in which at least one ascent begins at each vertex, except for the largest one, are counted by the Genocchi numbers of the first kind." - _Danny Rorabaugh_, Apr 25 2017

%D L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 49.

%D L. Euler, Institutionum Calculi Differentialis, volume 2 (1755), para. 181.

%D A. Fletcher, J. C. P. Miller, L. Rosenhead and L. J. Comrie, An Index of Mathematical Tables. Vols. 1 and 2, 2nd ed., Blackwell, Oxford and Addison-Wesley, Reading, MA, 1962, Vol. 1, p. 73.

%D A. Genocchi, Intorno all'espressione generale de'numeri Bernulliani, Ann. Sci. Mat. Fis., 3 (1852), 395-405.

%D R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics. Addison-Wesley, Reading, MA, 1990, p. 528.

%D N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

%D R. P. Stanley, Enumerative Combinatorics, Cambridge, Vol. 2, 1999; see Problem 5.8.

%D H. M. Terrill and E. M. Terrill, Tables of numbers related to the tangent coefficients, J. Franklin Inst., 239 (1945), 64-67.

%D G. Viennot, Interpretations combinatoires des nombres d'Euler et de Genocchi, Seminar on Number Theory, 1981/1982, No. 11, 94 pp., Univ. Bordeaux I, Talence, 1982.

%H T. D. Noe and Seiichi Manyama, <a href="/A001469/b001469.txt">Table of n, a(n) for n = 1..275</a> (first 100 terms from T. D. Noe)

%H F. Alayont and N. Krzywonos, <a href="http://faculty.gvsu.edu/alayontf/notes/rook_polynomials_higher_dimensions_preprint.pdf">Rook Polynomials in Three and Higher Dimensions</a>, 2012.

%H R. C. Archibald, <a href="http://dx.doi.org/10.1090/S0025-5718-45-99088-0">Review of Terrill-Terrill paper</a>, Math. Comp., 1 (1945), pp. 385-386.

%H R. B. Brent, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL18/Brent/brent5.html">Generalizing Tuenter's Binomial Sums</a>, J. Int. Seq. 18 (2015) # 15.3.2.

%H Alexander Burstein, Sergi Elizalde and Toufik Mansour, <a href="http://arXiv.org/abs/math.CO/0610234">Restricted Dumont permutations, Dyck paths and noncrossing partitions</a>, arXiv:math/0610234 [math.CO], 2006.

%H M. Domaratzki, <a href="http://www.cs.queensu.ca/TechReports/Reports/2001-449.ps">A Generalization of the Genocchi Numbers with Applications to ...</a>

%H M. Domaratzki, <a href="http://www.cs.uwaterloo.ca/journals/JIS/VOL7/Domaratzki/doma23.html">Combinatorial Interpretations of a Generalization of the Genocchi Numbers</a>, Journal of Integer Sequences, Vol. 7 (2004), Article 04.3.6.

%H D. Dumont, <a href="http://dx.doi.org/10.1016/0012-365X(72)90039-8">Sur une conjecture de Gandhi concernant les nombres de Genocchi</a>, (in French), Discrete Mathematics 1 (1972) 321-327.

%H D. Dumont, <a href="http://dx.doi.org/10.1215/S0012-7094-74-04134-9">Interprétations combinatoires des nombres de Genocchi</a>, Duke Math. J., 41 (1974), 305-318.

%H D. Dumont, <a href="/A001469/a001469_3.pdf">Interprétations combinatoires des nombres de Genocchi</a>, Duke Math. J., 41 (1974), 305-318. (Annotated scanned copy)

%H Dominique Dumont, Arthur Randrianarivony, <a href="http://dx.doi.org/10.1016/S0195-6698(95)90053-5">Sur une extension des nombres de Genocchi</a>, European J. Combin. 16 (1995), 147-151.

%H Dominique Dumont, Arthur Randrianarivony, <a href="http://dx.doi.org/10.1016/0012-365X(94)90230-5">Dérangements et nombres de Genocchi</a>, Discrete Math. 132 (1994), 37-49.

%H Richard Ehrenborg & 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 J. M. Gandhi, <a href="http://www.jstor.org/stable/2317385">Research Problems: A Conjectured Representation of Genocchi Numbers</a>, Amer. Math. Monthly 77 (1970), no. 5, 505-506. MR1535914.

%H I. M. Gessel, <a href="http://www.arXiv.org/abs/math.CO/0108121">Applications of the classical umbral calculus</a>, arXiv:math/0108121 [math.CO], 2001.

%H J. M. Hammersley, <a href="http://www.appliedprobability.org/data/files/TMS%20articles/14_1_1.pdf">An undergraduate exercise in manipulation</a>, Math. Scientist, 14 (1989), 1-23.

%H J. M. Hammersley, <a href="/A006846/a006846.pdf">An undergraduate exercise in manipulation</a>, Math. Scientist, 14 (1989), 1-23. (Annotated scanned copy)

%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.

%H D. H. Lehmer, <a href="http://www.jstor.org/stable/1968647">Lacunary recurrence formulas for the numbers of Bernoulli and Euler</a>, Annals Math., 36 (1935), 637-649.

%H H. Liang and Wuyungaowa, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL15/Liang/liang2.html">Identities Involving Generalized Harmonic Numbers and Other Special Combinatorial Sequences</a>, J. Int. Seq. 15 (2012) #12.9.6

%H Qui-Ming Luo, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL12/Luo/luo6.html">Fourier expansions and integral representations for Genocchi Polynomials</a>, JIS 12 (2009) 09.1.4.

%H T. Mansour, <a href="http://arXiv.org/abs/math.CO/0209379">Restricted 132-Dumont permutations</a>, arXiv:math/0209379 [math.CO], 2002.

%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...</a>, Adv. Appl. Math. 17 (1996), 1-26.

%H John Riordan & Paul R. Stein, <a href="http://dx.doi.org/10.1016/0012-365X(73)90131-3">Proof of a conjecture on Genocchi numbers</a>, Discrete Math. 5 (1973), 381-388. MR0316372 (47 #4919).

%H N. J. A. Sloane, <a href="/A001469/a001469_1.pdf">Rough notes on Genocchi numbers</a>

%H H. M. Terrill and E. M. Terrill, <a href="/A001469/a001469.pdf">Tables of numbers related to the tangent coefficients</a>, J. Franklin Inst., 239 (1945), 64-67. [Annotated scanned copy]

%H H. J. H. Tuenter, <a href="http://arXiv.org/abs/math.NT/0606080">Walking into an absolute sum</a>, arXiv:math/0606080 [math.NT], 2006.

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/GenocchiNumber.html">Genocchi Number.</a>

%H J. Worpitsky, <a href="/A001469/a001469_2.pdf">Studien ueber die Bernoullischen und Eulerschen Zahlen</a>, Journal für die reine undangewandte Mathematik (Crelle), 94 (1883), 203-232. See page 232. [Annotated scanned copy]

%H <a href="/index/Be#Bernoulli">Index entries for sequences related to Bernoulli numbers.</a>

%F a(n) = 2*(1-4^n)*B_{2n} (B = Bernoulli numbers).

%F x*tan(x/2) = Sum_{n>=1} x^(2*n)*abs(a(n))/(2*n)! = x^2/2 + x^4/24 + x^6/240 + 17*x^8/40320 + 31*x^10/725760 + O(x^11).

%F E.g.f.: 2*x/(1 + exp(x)) = x + Sum_{n>=1} a(2*n)*x^(2*n)/(2*n)! = -x^2/2! + x^4/4! - 3 x^6/6! + 17 x^8/8! + ...

%F O.g.f.: Sum_{n>=0} n!^2*(-x)^(n+1) / Product_{k=1..n} (1-k^2*x). - _Paul D. Hanna_, Jul 21 2011

%F a(n) = Sum_{k=0..2n-1} 2^k*B(k)*binomial(2*n,k) where B(k) is the k-th Bernoulli number. - _Benoit Cloitre_, May 31 2003

%F abs(a(n)) = Sum_{k=0..2n} (-1)^(n-k+1)*Stirling2(2n, k)*A059371(k). - _Vladeta Jovovic_, Feb 07 2004

%F G.f.: -x/(1+x/(1+2x/(1+4x/(1+6x/(1+9x/(1+12x/(1+16x/(1+20x/(1+25x/(1+...(continued fraction). - _Philippe Deléham_, Nov 22 2011

%F E.g.f.: E(x) = 2*x/(exp(x)+1) = x*(1-(x^3+2*x^2)/(2*G(0)-x^3-2*x^2)); G(k) = 8*k^3 + (12+4*x)*k^2 + (4+6*x+2*x^2)*k + x^3 + 2*x^2 + 2*x - 2*(x^2)*(k+1)*(2*k+1)*(x+2*k)*(x+2*k+4)/G(k+1); (continued fraction, Euler's kind, 1-step). - _Sergei N. Gladkovskii_, Jan 18 2012

%F a(n) = (-1)^n*(2*n)!*Pi^(-2*n)*4*(1-4^(-n))*Li{2*n}(1). - _Peter Luschny_, Jun 29 2012

%F Asymptotic: abs(a(n)) ~ 8*Pi*(2^(2*n)-1)*(n/(Pi*exp(1)))^(2*n+1/2)*exp(1/2+(1/24)/n-(1/2880)/n^3+(1/40320)/n^5+...). - _Peter Luschny_, Jul 24 2013

%F G.f.: x/(T(0)-x) -1, where T(k) = 2*x*k^2 + 4*x*k + 2*x - 1 - x*(-1+x+2*x*k+x*k^2)*(k+2)^2/T(k+1); (continued fraction). - _Sergei N. Gladkovskii_, Nov 17 2013

%F G.f.: -1 + x/(T(0)+x), where T(k) = 1 + (k+1)*(k+2)*x/(1+x*(k+2)^2/T(k+1)); (continued fraction). - _Sergei N. Gladkovskii_, Nov 17 2013

%p A001469 := proc(n::integer) (2*n)!*coeftayl( 2*x/(exp(x)+1), x=0,2*n) end proc:

%p for n from 1 to 20 do print(A001469(n)) od : # _R. J. Mathar_, Jun 22 2006

%t a[n_] := 2*(1-4^n)*BernoulliB[2n]; Table[a[n], {n, 17}] (* _Jean-François Alcover_, Nov 24 2011 *)

%t a[n_] := 2*n*EulerE[2*n-1, 0]; Table[a[n], {n, 17}] (* _Jean-François Alcover_, Jul 02 2013 *)

%o (PARI) a(n)=if(n<1,0,n*=2; 2*(1-2^n)*bernfrac(n))

%o (PARI) {a(n)=polcoeff(sum(m=0, n, m!^2*(-x)^(m+1)/prod(k=1, m, 1-k^2*x+x*O(x^n))), n)} /* _Paul D. Hanna_, Jul 21 2011 */

%o (Sage) # Algorithm of L. Seidel (1877)

%o # n -> [a(1), ..., a(n)] for n >= 1.

%o def A001469_list(n) :

%o D = [0]*(n+2); D[1] = -1

%o R = []; 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 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[h])

%o return R

%o A001469_list(17) # _Peter Luschny_, Jun 29 2012

%Y Cf. A110501, A000182, A006846, A012509, A226158.

%Y a(n) = -A065547(n, 1) and A065547(n+1, 2) for n >= 1.

%K sign,easy,nice

%O 1,3

%A _N. J. A. Sloane_

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Last modified August 21 11:52 EDT 2018. Contains 313939 sequences. (Running on oeis4.)