|
|
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. - From N. J. A. Sloane, Jan 02 2013
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) - From N. J. A. Sloane, Jun 12 2012
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.
|
|
|
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). [From Paul D. Hanna, Jul 21 2011]
a(n)=sum(k=0, 2n-1, 2^k*B(k)*C(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). - From 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
|
