|
%I #149 Mar 05 2024 02:30:01
%S 1,-1,0,1,0,-3,0,17,0,-155,0,2073,0,-38227,0,929569,0,-28820619,0,
%T 1109652905,0,-51943281731,0,2905151042481,0,-191329672483963,0,
%U 14655626154768697,0,-1291885088448017715,0,129848163681107301953
%N Genocchi numbers (of first kind): expansion of 2*x/(exp(x)+1).
%C The sign of a(1) depends on which convention one chooses: B(n) = B_n(1) or B(n) = B_n(0) where B(n) are the Bernoulli numbers and B_n(x) the Bernoulli polynomials (see the Wikipedia article on Bernoulli numbers). The definition given is in line with B(n) = B_n(0). The convention B(n) = B_n(1) corresponds to the e.g.f. -2*x/(1+exp(-x)). - _Peter Luschny_, Jun 28 2013
%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
%C Named after the Italian mathematician Angelo Genocchi (1817-1889). - _Amiram Eldar_, Jun 06 2021
%C Conjecture: For any positive integer n, -a(n+1) is the permanent of the n X n matrix M with M(j, k) = floor((2*j - k)/n), (j,k=1..n). - _Zhi-Wei Sun_, Sep 07 2021
%C A corresponding conjecture can also be made for L. Seidel's 'Genocchi numbers of second kind' A005439. - _Peter Luschny_, Sep 08 2021
%D Louis Comtet, Advanced Combinatorics, Reidel, 1974, p. 49.
%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 Ronald L. Graham, Donald E. Knuth and Oren Patashnik, Concrete Mathematics. Addison-Wesley, Reading, MA, 1990, p. 528.
%D Richard P. Stanley, Enumerative Combinatorics, Cambridge, Vol. 2, 1999; see Problem 5.8.
%H Seiichi Manyama, <a href="/A036968/b036968.txt">Table of n, a(n) for n = 1..551</a>
%H Shreya Ahirwar, Susanna Fishel, Parikshita Gya, Pamela E. Harris, Nguyen Pham, Andrés R. Vindas-Meléndez, and Dan Khanh Vo, <a href="https://doi.org/10.37236/10983">Maximal Chains in Bond Lattices</a>, Elec. J. Combinatorics (2022) Vol. 29, No. 3, #P3.11.
%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., Vol. 1, No. 10 (1945), pp. 385-386.
%H Beáta Bényi and Matthieu Josuat-Vergès, <a href="https://arxiv.org/abs/2010.10060">Combinatorial proof of an identity on Genocchi numbers</a>, arXiv:2010.10060 [math.CO], 2020.
%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), Article 03.4.8.
%H Dominique Dumont, <a href="http://projecteuclid.org/euclid.dmj/1077310398">Interpretations combinatoires des nombres de Genocchi</a>, Duke Math. J., Vol. 41 (1974), pp. 305-318.
%H Dominique Dumont, <a href="/A001469/a001469_3.pdf">Interprétations combinatoires des nombres de Genocchi</a>, Duke Math. J., Vol. 41 (1974), pp. 305-318. (Annotated scanned copy)
%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 Ira M. Gessel, <a href="https://doi.org/10.5281/zenodo.7625111">On the Almkvist-Meurman Theorem for Bernoulli Polynomials</a>, Integers (2023) Vol. 23, #A14.
%H Hans-Christian Herbig, Daniel Herden and Christopher Seaton, <a href="https://doi.org/10.1090/proc/12806">On compositions with x^2/(1-x)</a>, Proceedings of the American Mathematical Society, Vol. 143, No. 11 (2015), pp. 4583-4596; <a href="http://arxiv.org/abs/1404.1022">arXiv preprint</a>, arXiv:1404.1022 [math.SG], 2014.
%H Gábor Hetyei, <a href="https://doi.org/10.1016/j.ejc.2019.04.007">Alternation acyclic tournaments</a>, European Journal of Combinatorics, Vol. 81 (2019), pp. 1-21; <a href="https://arxiv.org/abs/1704.07245">arXiv preprint</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 (1997), pp. 49-58.
%H Ali Lavasani and Sagar Vijay, <a href="https://arxiv.org/abs/2402.14906">The Stability of Gapped Quantum Matter and Error-Correction with Adiabatic Noise</a>, arXiv:2402.14906 [cond-mat.str-el], 2024. See p. 16.
%H Chellal Redha, <a href="https://arxiv.org/abs/2402.17063">An Identity for Generalized Euler Polynomials</a>, arXiv:2402.17063 [math.NT], 2024. See p. 7.
%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 H. M. Terrill and E. M. Terrill, <a href="https://ur.booksc.eu/ireader/2106189">Tables of numbers related to the tangent coefficients</a>, J. Franklin Inst., 239 (1945), 66-67.
%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Bernoulli_number">Bernoulli number</a>.
%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Genocchi_number">Genocchi number</a>.
%F E.g.f.: 2*x/(exp(x)+1).
%F a(n) = 2*(1-2^n)*B_n (B = Bernoulli numbers). - _Benoit Cloitre_, Oct 26 2003
%F 2*x/(exp(x)+1) = x + Sum_{n>=1} x^(2*n)*G_{2*n}/(2*n)!.
%F a(n) = Sum_{k=0..n-1} binomial(n,k) 2^k*B(k). - _Peter Luschny_, Apr 30 2009
%F From _Sergei N. Gladkovskii_, Dec 12 2012 to Nov 23 2013: (Start) Continued fractions:
%F E.g.f.: 2*x/(exp(x)+1) = x - x^2/2*G(0) where G(k) = 1 - x^2/(x^2 + 4*(2*k+1)*(2*k+3)/G(k+1)).
%F E.g.f.: 2/(E(0)+1) where E(k) = 1 + x/(2*k+1 - x*(2*k+1)/(x + (2*k+2)/E(k+1))).
%F G.f.: 2 - 1/G(0) where G(k) = 1 - x*(k+1)/(1 + x*(k+1)/(1 - x*(k+1)/(1 + x*(k+1)/G(k+1)))).
%F E.g.f.: 2*x/(1 + exp(x)) = 2*x-2 - 2*T(0), where T(k) = 4*k-1 + x/(2 - x/( 4*k+1 + x/(2 - x/T(k+1)))).
%F G.f.: 2 - Q(0)/(1-x+x^2) where Q(k) = 1 - x^4*(k+1)^4/(x^4*(k+1)^4 - (1 - x + x^2 + 2*x^2*k*(k+1))*(1 - x + x^2 + 2*x^2*(k+1)*(k+2))/Q(k+1)). (End)
%F a(n) = n*zeta(1-n)*(2^(n+1)-2) for n > 1. - _Peter Luschny_, Jun 28 2013
%F O.g.f.: x*Sum_{n>=0} n! * (-x)^n / (1 - n*x) / Product_{k=1..n} (1 - k*x). - _Paul D. Hanna_, Aug 03 2014
%F Sum_{n>=1} 1/a(2*n) = A321595. - _Amiram Eldar_, May 07 2021
%F a(n) = (-1)^n*2*n*PolyLog(1 - n, -1). - _Peter Luschny_, Aug 17 2021
%p a := n -> n*euler(n-1,0); # _Peter Luschny_, Jul 13 2009
%t a[n_] := n*EulerE[n - 1, 0]; Table[a[n], {n, 1, 32}] (* _Jean-François Alcover_, Dec 08 2011, after _Peter Luschny_ *)
%t Range[0, 31]! CoefficientList[ Series[ 2x/(1 + Exp[x]), {x, 0, 32}], x] (* _Robert G. Wilson v_, Oct 26 2012 *)
%t Table[(-1)^n 2 n PolyLog[1 - n, -1], {n, 1, 32}] (* _Peter Luschny_, Aug 17 2021 *)
%o (PARI) {a(n) = if( n<0, 0, n! * polcoeff( 2*x / (1 + exp(x + x * O(x^n))), n))}; /* _Michael Somos_, Jul 23 2005 */
%o (PARI) /* From o.g.f. (_Paul D. Hanna_, Aug 03 2014): */
%o {a(n)=local(A=1); A=x*sum(m=0, n, m!*(-x)^m/(1-m*x)/prod(k=1,m,1 - k*x +x*O(x^n))); polcoeff(A, n)}
%o for(n=1, 32, print1(a(n), ", "))
%o (Sage) # with a(1) = -1
%o [z*zeta(1-z)*(2^(z+1)-2) for z in (1..32)] # _Peter Luschny_, Jun 28 2013
%o (Sage)
%o def A036968_list(len):
%o e, f, R, C = 4, 1, [], [1]+[0]*(len-1)
%o for n in (2..len-1):
%o for k in range(n, 0, -1):
%o C[k] = C[k-1] / (k+1)
%o C[0] = -sum(C[k] for k in (1..n))
%o R.append((2-e)*f*C[0])
%o f *= n; e *= 2
%o return R
%o print(A036968_list(34)) # _Peter Luschny_, Feb 22 2016
%o (Python)
%o from sympy import bernoulli
%o def A036968(n): return (2-(2<<n))*bernoulli(n) # _Chai Wah Wu_, Apr 14 2023
%Y A001469 is the main entry for this sequence. A226158 is another version.
%Y Cf. A005439 (Genocchi numbers of second kind).
%Y Cf. A083007, A083008, A083009, A083010, A083011, A083012, A083013, A083014, A321595.
%K sign,easy,nice
%O 1,6
%A _N. J. A. Sloane_
|
