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A036968
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Genocchi numbers (of first kind): expansion of 2*x/(exp(x)+1).
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27
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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
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OFFSET
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1,6
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COMMENTS
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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
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REFERENCES
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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.
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LINKS
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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
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FORMULA
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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
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MAPLE
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a := n -> n*euler(n-1, 0); # Peter Luschny, Jul 13 2009
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MATHEMATICA
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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 *)
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PROG
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(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), ", "))
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CROSSREFS
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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 A009759
Adjacent sequences: A036965 A036966 A036967 * A036969 A036970 A036971
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KEYWORD
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sign,easy,nice,changed
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AUTHOR
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N. J. A. Sloane
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STATUS
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approved
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