|
| |
|
|
A005439
|
|
Genocchi medians (or Genocchi numbers of second kind).
(Formerly M1888)
|
|
8
| |
|
|
1, 2, 8, 56, 608, 9440, 198272, 5410688, 186043904, 7867739648, 401293838336, 24290513745920, 1721379917619200, 141174819474169856, 13266093250285568000, 1415974941618255921152, 170361620874699124637696
(list; graph; refs; listen; history; internal format)
|
|
|
|
OFFSET
| 1,2
|
|
|
COMMENTS
| 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
Hankel transform of 1,1,2,8,... is A168488. [From Paul Barry (pbarry(AT)wit.ie), Nov 27 2009]
|
|
|
REFERENCES
| 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.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
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.
|
|
|
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. [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.
|
|
|
FORMULA
| 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
G.f.: (of 1,1,2,8,...) 1/(1-x-x^2/(1-5x-16x^2/(1-13x-81x^2/(1-25x-256x^2/(1-41x-625x^2/(1-... (continued fraction). [From Paul Barry (pbarry(AT)wit.ie), Nov 27 2009]
G.f.: A(x) = 1/S(0), S(k) = 1 - x*(k+1)*(k+2)/(1 - x*(k+1)*(k+2)/S(k+1)) (continued fraction). - Sergei N. Gladkovskii, Dec 14 2011
G.f.: A(x) = -1/S(0), S(k) = 2*x*(k+1)^2 - 1 - x^2*(k+1)^2*(k+2)^2/S(k+1) (continued fraction). - Sergei N. Gladkovskii, Dec 14 2011
|
|
|
MATHEMATICA
| a[n_] = 2*(-1)^(n-2)* Sum[Binomial[n, k]*(1 - 2^(n+k+1))*BernoulliB[n+k+1], {k, 0, n}]; Table[a[n], {n, 16}] (* From Jean-François Alcover, Jul 18 2011, after PARI prog. *)
|
|
|
PROG
| (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)
|
|
|
CROSSREFS
| a(n) = A000366(n)*2^(n-1). See A000366 for further information.
Sequence in context: A097691 A124212 A197949 * A128814 A108208 A203199
Adjacent sequences: A005436 A005437 A005438 * A005440 A005441 A005442
|
|
|
KEYWORD
| nonn,nice,easy
|
|
|
AUTHOR
| Simon Plouffe (simon.plouffe(AT)gmail.com)
|
|
|
EXTENSIONS
| More terms and additional comments from David W. Wilson (davidwwilson(AT)comcast.net), Jan 11 2001
|
| |
|
|
