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 A036968 Genocchi numbers (of first kind): expansion of 2*x/(exp(x)+1). 27
 1, -1, 0, 1, 0, -3, 0, 17, 0, -155, 0, 2073, 0, -38227, 0, 929569, 0, -28820619, 0, 1109652905, 0, -51943281731, 0, 2905151042481, 0, -191329672483963, 0, 14655626154768697, 0, -1291885088448017715, 0, 129848163681107301953 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,6 COMMENTS 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 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 REFERENCES L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 49. 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. R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics. Addison-Wesley, Reading, MA, 1990, p. 528. R. P. Stanley, Enumerative Combinatorics, Cambridge, Vol. 2, 1999; see Problem 5.8. H. M. Terrill and E. M. Terrill, Tables of numbers related to the tangent coefficients, J. Franklin Inst., 239 (1945), 66-67. LINKS Seiichi Manyama, Table of n, a(n) for n = 1..551 R. C. Archibald, Review of Terrill-Terrill paper, Math. Comp., 1 (1945), pp. 385-386. Beáta Bényi and Matthieu Josuat-Vergès, Combinatorial proof of an identity on Genocchi numbers, arXiv:2010.10060 [math.CO], 2020. Kwang-Wu Chen, An Interesting Lemma for Regular C-fractions, J. Integer Seqs., Vol. 6, 2003. D. Dumont, Interpretations combinatoires des nombres de Genocchi, Duke Math. J., 41 (1974), 305-318. D. Dumont, Interprétations combinatoires des nombres de Genocchi, Duke Math. J., 41 (1974), 305-318. (Annotated scanned copy) H.-C. Herbig, D. Herden, C. Seaton, On compositions with x^2/(1-x), arXiv preprint arXiv:1404.1022 [math.SG], 2014. Gábor Hetyei, Alternation acyclic tournaments, arXiv:math/1704.07245 [math.CO], 2017. G. Kreweras, Sur les permutations comptées par les nombres de Genocchi de 1-ière et 2-ième espèce, Europ. J. Comb., vol. 18, pp. 49-58, 1997. Wikipedia, Bernoulli number Wikipedia, Genocchi number FORMULA E.g.f.: 2*x/(exp(x)+1). a(n) = 2*(1-2^n)*B_n (B = Bernoulli numbers). - Benoit Cloitre, Oct 26 2003 2*x/(exp(x)+1) = x + Sum_{n>0} x^(2*n)*G_{2*n}/(2*n)!. a(n) = Sum_{k=0^(n-1)} binomial(n,k) 2^k*B(k). - Peter Luschny, Apr 30 2009 From Sergei N. Gladkovskii, Dec 12 2012 to Nov 23 2013: (Start): Continued fractions: 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)). 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))). 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)))). 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)))). 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) a(n) = n*zeta(1-n)*(2^(n+1)-2) for n > 1. - Peter Luschny, Jun 28 2013 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 MAPLE a := n -> n*euler(n-1, 0); # Peter Luschny, Jul 13 2009 MATHEMATICA a[n_] := n*EulerE[n - 1, 0]; Table[a[n], {n, 1, 32}] (* Jean-François Alcover, Dec 08 2011, after Peter Luschny *) Range[0, 31]! CoefficientList[ Series[ 2x/(1 + Exp[x]), {x, 0, 32}], x] (* Robert G. Wilson v, Oct 26 2012 *) PROG (PARI) {a(n) = if( n<0, 0, n! * polcoeff( 2*x / (1 + exp(x + x * O(x^n))), n))}; /* Michael Somos, Jul 23 2005 */ (Sage) # with a(1) = -1 [z*zeta(1-z)*(2^(z+1)-2) for z in (1..32)]  # Peter Luschny, Jun 28 2013 # Alternatively: def A036968_list(len):     e, f, R, C = 4, 1, [], [1]+[0]*(len-1)     for n in (2..len-1):         for k in range(n, 0, -1):             C[k] = C[k-1] / (k+1)         C[0] = -sum(C[k] for k in (1..n))         R.append((2-e)*f*C[0])         f *= n; e *= 2     return R print(A036968_list(34)) # Peter Luschny, Feb 22 2016 (PARI) /* From o.g.f. (Paul D. Hanna, Aug 03 2014): */ {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)} for(n=1, 32, print1(a(n), ", ")) CROSSREFS A001469 is the main entry for this sequence. A226158 is another version. Cf. A083007, A083008, A083009, A083010, A083011, A083012, A083013, A083014. Sequence in context: A038122 A143779 A240244 * A226158 A024040 A338489 Adjacent sequences:  A036965 A036966 A036967 * A036969 A036970 A036971 KEYWORD sign,easy,nice AUTHOR STATUS approved

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Last modified April 13 06:52 EDT 2021. Contains 342935 sequences. (Running on oeis4.)