1,3

The Genocchi numbers satisfy Seidel's recurrence: for n>1, 0 = sum{j=0..[n/2], 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

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

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

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.

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.

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.

H. M. Terrill and E. M. Terrill, Tables of numbers related to the tangent coefficients, J. Franklin Inst., 239 (1945), 64-67.

R. P. Stanley, Enumerative Combinatorics, Cambridge, Vol. 2, 1999; see Problem 5.8.

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.

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 des 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, Link to a section of The World of Mathematics.

Index entries for sequences related to Bernoulli numbers.

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

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

2x/(1 + e^x) = x + Sum_{n >= 1} a(2n) x^(2n) / (2n)! = - x^2/2! + x^4/4! - 3 x^6/6! + 17 x^8/8! + ...

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 (benoit7848c(AT)orange.fr), May 31 2003

|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

A001469 := proc(n::integer) RETURN( (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 (mathar(AT)strw.leidenuniv.nl), Jun 22 2006

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

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

Sequence in context: A145081 A020562 A135751 this_sequence A110501 A066211 A163884

Adjacent sequences: A001466 A001467 A001468 this_sequence A001470 A001471 A001472

sign,easy,nice

N. J. A. Sloane (njas(AT)research.att.com).

More terms from James A. Sellers (sellersj(AT)math.psu.edu), Jul 06 2000