1,3

The Genocchi numbers satisfy Seidel's recurrence: for n>1, 0 = sum{j=0..[n/2], (-1)^j*C(n,2j)*a(n-j)}. - R. 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. - R. Stephan, Apr 26 2004

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

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

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

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

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

R. P. Stanley, Enumerative Combinatorics, Cambridge, Vol. 2, 1999; p. 74 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.

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

A002105(n) = 2^(n-1)/n * a(n). - D. E. Knuth, Jan 16 2007

E.g.f.: x tan(x/2) = Sum_{k > 0} a(k) x^(2k)/(2k)!.

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

O.g.f.: A(x) = x/(1-x/(1-2*x/(1-4*x/(1-6*x/(1-9*x/(1-12*x/(... -[(n+1)/2]*[(n+2)/2]*x/(1- ...)))))))) (continued fraction). - Paul D. Hanna (pauldhanna(AT)juno.com), Jan 16 2006

(PARI) a(n)=if(n<1, 0, 2*(-1)^n*(1-4^n)*bernfrac(2*n))

(PARI) {a(n)=if(n<1, 0, (2*n)!*polcoeff( x*tan(x/2+x*O(x^(2*n))), 2*n))}

Cf. A036968(2n)=A001469(n)=(-1)^n a(n).

Sequence in context: A020562 A135751 A001469 this_sequence A066211 A163884 A052143

Adjacent sequences: A110498 A110499 A110500 this_sequence A110502 A110503 A110504

nonn

Michael Somos, Jul 23 2005