1,2

a(n) is the number of Boolean functions of n variables whose ROBDD (reduced ordered binary decision diagram) contains exactly n branch nodes, one for each variable. - D. E. Knuth, Jul 11 2007

The earliest known reference for these numbers is Seidel (1877). - D. E. Knuth, Jul 13 2007

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

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

G. Han and J. Zeng, "On a q-sequence that generalizes the median Genocchi numbers", Annal Sci. Math. Quebec, 23(1999), no. 1, 63-72

G. Kreweras, Sur les permutations compte'es par les nombres de Genocchi..., Europ. J. Comb., vol. 18, pp. 49-58, 1997. (See also page 76.)

L. Seidel, Ueber eine einfache Entstehungsweise der Bernoulli'schen Zahlen und einiger verwandten Reihen, Sitzungsberichte der mathematisch-physikalischen Classe der k\"oniglich bayerischen Akademie der Wissenschaften zu M\"unchen, volume 7 (1877), 157-187.

G. Viennot, Interpretations combinatoires des nombres d'Euler et de Genocchi. Seminar on Number Theory, 1981/1982, Exp. 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. [Theorem 3.5]

Kwang-Wu Chen, An Interesting Lemma for Regular C-fractions, J. Integer Seqs., Vol. 6, 2003.

D. Dumont and J. Zeng, Polynomes d'Euler et les fractions continues de Stieltjes-Rogers, Ramanujan J. 2 (1998) 3, 387-410.

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

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

a(n) = T(n, 1) where T(1, x) = 1; T(n, x) = (x+1)*((x+1)T(n-1, x+1)-xT(n-1, x)); see A058942.

2 * (-1)^n * sum{k=0..n, C(n, k)*(1-2^(n+k+1))*B(n+k+1)}, with B(n) the Bernoulli numbers. - R. Stephan, Apr 17 2004

O.g.f.: 1 + x*A(x) = 1/(1-x/(1-x/(1-4*x/(1-4*x/(1-9*x/(1-9*x/(... -[(n+1)/2]^2*x/(1- ...)))))))) (continued fraction). - Paul D. Hanna (pauldhanna(AT)juno.com), Oct 07 2005

(PARI) a(n)=2*(-1)^n*sum(k=0, n, binomial(n, k)*(1-2^(n+k+1))*bernfrac(n+k+1))

(PARI) {a(n)=local(CF=1+x*O(x^(n+2))); if(n<0, return(0), for(k=1, n+1, CF=1/(1-((n-k+1)\2+1)^2*x*CF)); return(Vec(CF)[n+2]))} (Hanna)

a(n) = A000366(n)*2^(n-1). See A000366 for further information.

Sequence in context: A113248 A097691 A124212 this_sequence A128814 A108208 A135079

Adjacent sequences: A005436 A005437 A005438 this_sequence A005440 A005441 A005442

nonn,nice,easy

Simon Plouffe (simon.plouffe(AT)gmail.com)

More terms and additional comments from David W. Wilson (davidwwilson(AT)comcast.net), Jan 11 2001