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A036968 Genocchi numbers (of first kind): expansion of 2*x/(exp(x)+1). 18
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

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

Table of n, a(n) for n=1..32.

R. C. Archibald, Review of Terrill-Terrill paper, Math. Comp., 1 (1945), pp. 385-386.

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.

H.-C. Herbig, D. Herden, C. Seaton, On compositions with x^2/(1-x), arXiv preprint arXiv:1404.1022, 2014

G. Kreweras, Sur les permutations comptées par les nombres de Genocchi de 1-ière et 2-ième sspèce, Europ. J. Comb., vol. 18, pp. 49-58, 1997.

Wikipedia, Bernoulli number

Wikipedia, Genocchi number

FORMULA

E.g.f.: 2x/(exp(x)+1).

a(n) = 2*(1-2^n)*B_n (B = Bernoulli numbers). - Benoit Cloitre, Oct 26 2003

2x/(exp(x)+1) = x + Sum_{n>0} x^(2n)*G_{2n}/(2n)!.

a(n) = Sum_{k=0^(n-1)} binomial(n,k) 2^k B(k). - Peter Luschny, Apr 30 2009

E.g.f.: 2x/(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) ); (recursively defined continued fraction). - Sergei N. Gladkovskii, Dec 12 2012

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) )); (recursively defined continued fraction). - Sergei N. Gladkovskii, Jan 09 2013.

a(n) = n*zeta(1-n)*(2^(n+1)-2) for n>1. - Peter Luschny, Jun 28 2013

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) ))); (continued fraction). - Sergei N. Gladkovskii, Jul 07 2013

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) ))); (continued fraction). - Sergei N. Gladkovskii, Nov 23 2013

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) ); (continued fraction). - Sergei N. Gladkovskii, Nov 23 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

(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: A143779 A240244 * A226158 A024040 A009759 A127187

Adjacent sequences:  A036965 A036966 A036967 * A036969 A036970 A036971

KEYWORD

sign,easy,nice

AUTHOR

N. J. A. Sloane.

STATUS

approved

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Last modified August 3 23:27 EDT 2015. Contains 260265 sequences.