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A001469 Genocchi numbers (of first kind); unsigned coefficients give expansion of x*tan(x/2).
(Formerly M3041 N1233)
44
-1, 1, -3, 17, -155, 2073, -38227, 929569, -28820619, 1109652905, -51943281731, 2905151042481, -191329672483963, 14655626154768697, -1291885088448017715, 129848163681107301953, -14761446733784164001387 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,3

COMMENTS

The Genocchi numbers satisfy Seidel's recurrence: for n>1, 0 = sum{j=0..[n/2], C(n,2j)*a(n-j)}. - Ralf Stephan, Apr 17 2004

The (n+1)st Genocchi number is the number of Dumont permutations of the first kind on 2n letters. In a Dumont permutation of first kind, each even integer must be followed by a smaller integer and each odd integer is either followed by a larger integer or is the last element. - Ralf Stephan, Apr 26 2004

REFERENCES

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

F. Alayont and N. Krzywonos, Rook Polynomials in Three and Higher Dimensions, http://faculty.gvsu.edu/alayontf/notes/rook_polynomials_higher_dimensions_preprint.pdf. 2012.

L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 49.

Dumont, Dominique. Sur une conjecture de Gandhi concernant les nombres de Genocchi. (French) Discrete Math. 1 (1972), no. 4, 321--327. MR0296012 (45 #5073)

D. Dumont, Interpretations combinatoires des nombres de Genocchi, Duke Math. J., 41 (1974), 305-318.

Dumont, Dominique and Randrianarivony, Arthur, Sur une extension des nombres de Genocchi, European J. Combin. 16 (1995), 147-151.

Dumont, Dominique and Randrianarivony, Arthur, Derangements et nombres de Genocchi, Discrete Math. 132 (1994), 37-49.

R. Ehrenborg and E. Steingrimsson, Yet another triangle for the Genocchi numbers, Europ. J. Combin., 21 (2000), 593-600.

L. Euler, Institutionum Calculi Differentialis, volume 2 (1755), para. 181.

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.

Gandhi, J. M. Research Problems: A Conjectured Representation of Genocchi Numbers. Amer. Math. Monthly 77 (1970), no. 5, 505--506.MR1535914

A. Genocchi, Intorno all'espressione generale de'numeri Bernulliani, Ann. Sci. Mat. Fis., 3 (1852), 395-405.

R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics. Addison-Wesley, Reading, MA, 1990, p. 528.

J. M. Hammersley, An undergraduate exercise in manipulation, Math. Scientist, 14 (1989), 1-23.

G. Kreweras, Sur les permutations compte'es par les nombres de Genocchi..., Europ. J. Comb., vol. 18, pp. 49-58, 1997.

D. H. Lehmer, Lacunary recurrence formulas for the numbers of Bernoulli and Euler, Annals Math., 36 (1935), 637-649.

Riordan, John; Stein, Paul R. Proof of a conjecture on Genocchi numbers. Discrete Math. 5 (1973), 381--388. MR0316372 (47 #4919).

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

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), 64-67.

G. Viennot, Interpretations combinatoires des nombres d'Euler et de Genocchi, Seminar on Number Theory, 1981/1982, No. 11, 94 pp., Univ. Bordeaux I, Talence, 1982.

LINKS

T. D. Noe, Table of n, a(n) for n=1..100

Alexander Burstein, Sergi Elizalde and Toufik Mansour, Restricted Dumont permutations, Dyck paths and noncrossing partitions, arXiv math.CO/0610234.

M. Domaratzki, A Generalization of the Genocchi Numbers with Applications to ...

M. Domaratzki, Combinatorial Interpretations of a Generalization of the Genocchi Numbers, Journal of Integer Sequences, Vol. 7 (2004), Article 04.3.6.

I. M. Gessel, Applications of the classical umbral calculus.

T. Mansour, Restricted 132-Dumont permutations.

A. Randrianarivony and J. Zeng, Une famille de polynomes qui interpole plusieurs suites..., Adv. Appl. Math. 17 (1996), 1-26.

H. J. H. Tuenter, Walking into an absolute sum

Eric Weisstein's World of Mathematics, Genocchi Number.

Index entries for sequences related to Bernoulli numbers.

FORMULA

a(n) = 2*(1-4^n)*B_{2n} (B = Bernoulli numbers).

x*tan(x/2) = Sum_{n>=1} x^(2*n)*abs(a(n))/(2*n)! = x^2/2 + x^4/24 + x^6/240 + 17*x^8/40320 + 31*x^10/725760 + O(x^11).

E.g.f.: 2*x/(1 + exp(x)) = x + Sum_{n >= 1} a(2*n)*x^(2*n)/(2*n)! = - x^2/2! + x^4/4! - 3 x^6/6! + 17 x^8/8! + ...

O.g.f.: Sum_{n>=0} n!^2 * (-x)^(n+1) / Product_{k=1..n} (1 - k^2*x). [Paul D. Hanna, Jul 21 2011]

a(n) = sum(k=0..2n-1, 2^k*B(k)*C(2*n,k)) where B(k) is the k-th Bernoulli number and C(n,k) = binomial(n,k). - Benoit Cloitre, May 31 2003

abs(a(n)) = Sum_{k=0..2*n} (-1)^(n-k+1)*Stirling2(2*n, k)*A059371(k). - Vladeta Jovovic, Feb 07 2004

G.f.: -x/(1+x/(1+2x/(1+4x/(1+6x/(1+9x/(1+12x/(1+16x/(1+20x/(1+25x/(1+...(continued fraction). - Philippe Deléham, Nov 22 2011

E.g.f.: E(x)=2*x/(exp(x)+1)=x*(1-(x^3+2*x^2)/(2*G(0)-x^3-2*x^2)) ; G(k)= 8*k^3 + (12+4*x)*k^2 + (4+6*x+2*x^2)*k + x^3+2*x^2+2*x - 2*(x^2)*(k+1)*(2*k+1)*(x+2*k)*(x+2*k+4)/G(k+1); (continued fraction, Euler's kind, 1-step). - Sergei N. Gladkovskii, Jan 18 2012

a(n) = (-1)^n*(2*n)!*Pi^(-2*n)*4*(1-4^(-n))*Li{2*n}(1). - Peter Luschny, Jun 29 2012

Asymptotic: abs(a(n)) ~ 8*Pi*(2^(2*n)-1)*(n/(Pi*exp(1)))^(2*n+1/2)*exp(1/2+(1/24)/n-(1/2880)/n^3+(1/40320)/n^5+...). - Peter Luschny, Jul 24 2013

G.f.: x/(T(0)-x) -1, where T(k) = 2*x*k^2 + 4*x*k + 2*x -1 - x*(-1+x+2*x*k+x*k^2)*(k+2)^2/T(k+1) ; (continued fraction). - Sergei N. Gladkovskii, Nov 17 2013

G.f.: -1 + x/(T(0)+x), where T(k) = 1 + (k+1)*(k+2)*x/(1 + x*(k+2)^2/T(k+1) ); (continued fraction). - Sergei N. Gladkovskii, Nov 17 2013

MAPLE

A001469 := proc(n::integer) (2*n)!*coeftayl( 2*x/(exp(x)+1), x=0, 2*n) end: for n from 1 to 20 do print(A001469(n)) od : # R. J. Mathar, Jun 22 2006

MATHEMATICA

a[n_] := 2*(1-4^n)*BernoulliB[2n]; Table[a[n], {n, 1, 17}] (* Jean-François Alcover, Nov 24 2011 *)

a[n_] := 2*n*EulerE[2*n-1, 0]; Table[a[n], {n, 1, 17}] (* Jean-François Alcover, Jul 02 2013 *)

PROG

(PARI) a(n)=if(n<1, 0, n*=2; 2*(1-2^n)*bernfrac(n))

(PARI) {a(n)=polcoeff(sum(m=0, n, m!^2*(-x)^(m+1)/prod(k=1, m, 1-k^2*x+x*O(x^n))), n)} /* Paul D. Hanna, Jul 21 2011 */

(Sage) # Algorithm of L. Seidel (1877)

# n -> [a(1), ..., a(n)] for n >= 1.

def A001469_list(n) :

    D = [0]*(n+2); D[1] = -1

    R = []; b = False

    for i in(0..2*n-1) :

        h = i//2 + 1

        if b :

            for k in range(h-1, 0, -1) : D[k] -= D[k+1]

        else :

            for k in range(1, h+1, 1) :  D[k] -= D[k-1]

        b = not b

        if not b : R.append(D[h])

    return R

A001469_list(17) # Peter Luschny, Jun 29 2012

CROSSREFS

Cf. A000182, A006846. a(n)=-A065547(n, 1) and A065547(n+1, 2), n>=1.

Cf. A226158.

Sequence in context: A208832 A135751 A168441 * A110501 A066211 A163884

Adjacent sequences:  A001466 A001467 A001468 * A001470 A001471 A001472

KEYWORD

sign,easy,nice

AUTHOR

N. J. A. Sloane.

EXTENSIONS

More terms from James A. Sellers, Jul 06 2000

STATUS

approved

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Last modified November 27 08:10 EST 2014. Contains 250159 sequences.