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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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31
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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
Named after the Italian mathematician Angelo Genocchi (1817-1889). - Amiram Eldar, Jun 06 2021
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
A corresponding conjecture can also be made for L. Seidel's 'Genocchi numbers of second kind' A005439. - Peter Luschny, Sep 08 2021
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REFERENCES
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Louis 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.
Ronald L. Graham, Donald E. Knuth and Oren Patashnik, Concrete Mathematics. Addison-Wesley, Reading, MA, 1990, p. 528.
Richard P. Stanley, Enumerative Combinatorics, Cambridge, Vol. 2, 1999; see Problem 5.8.
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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., Vol. 1, No. 10 (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), Article 03.4.8.
Dominique Dumont, Interpretations combinatoires des nombres de Genocchi, Duke Math. J., Vol. 41 (1974), pp. 305-318.
Dominique Dumont, Interprétations combinatoires des nombres de Genocchi, Duke Math. J., Vol. 41 (1974), pp. 305-318. (Annotated scanned copy)
Shishuo Fu, Zhicong Lin and Zhi-Wei Sun, Proofs of five conjectures relating permanents to combinatorial sequences, arXiv:2109.11506 [math.CO], 2021.
Hans-Christian Herbig, Daniel Herden and Christopher Seaton, On compositions with x^2/(1-x), Proceedings of the American Mathematical Society, Vol. 143, No. 11 (2015), pp. 4583-4596; arXiv preprint, arXiv:1404.1022 [math.SG], 2014.
Gábor Hetyei, Alternation acyclic tournaments, European Journal of Combinatorics, Vol. 81 (2019), pp. 1-21; arXiv preprint, 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 (1997), pp. 49-58.
Zhi-Wei Sun, Arithmetic properties of some permanents, arXiv:2108.07723 [math.GM], 2021.
H. M. Terrill and E. M. Terrill, Tables of numbers related to the tangent coefficients, J. Franklin Inst., 239 (1945), 66-67.
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>=1} 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
Sum_{n>=1} 1/a(2*n) = A321595. - Amiram Eldar, May 07 2021
a(n) = (-1)^n*2*n*PolyLog(1 - n, -1). - Peter Luschny, Aug 17 2021
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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 *)
Table[(-1)^n 2 n PolyLog[1 - n, -1], {n, 1, 32}] (* Peter Luschny, Aug 17 2021 *)
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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 */
(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), ", "))
(Sage) # with a(1) = -1
[z*zeta(1-z)*(2^(z+1)-2) for z in (1..32)] # Peter Luschny, Jun 28 2013
(Sage)
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
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CROSSREFS
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A001469 is the main entry for this sequence. A226158 is another version.
Cf. A005439 (Genocchi numbers of second kind).
Cf. A083007, A083008, A083009, A083010, A083011, A083012, A083013, A083014, A321595.
Sequence in context: A038122 A143779 A240244 * A226158 A024040 A338489
Adjacent sequences: A036965 A036966 A036967 * A036969 A036970 A036971
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KEYWORD
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sign,easy,nice
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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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