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A005439 Genocchi medians (or Genocchi numbers of second kind).
(Formerly M1888)
20
1, 2, 8, 56, 608, 9440, 198272, 5410688, 186043904, 7867739648, 401293838336, 24290513745920, 1721379917619200, 141174819474169856, 13266093250285568000, 1415974941618255921152, 170361620874699124637696 (list; graph; refs; listen; history; text; 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. - Don Knuth, Jul 11 2007

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

Hankel transform of 1,1,2,8,... is A168488. - Paul Barry, Nov 27 2009

According to Hetyei [2017], alternation acyclic tournaments "are counted by the median Genocchi numbers"; an alternation acyclic tournament "does not contain a cycle in which descents and ascents alternate." - Danny Rorabaugh, Apr 25 2017

REFERENCES

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

LINKS

Muniru A Asiru, Table of n, a(n) for n = 1..270 (terms n = 1..100 from T. D. Noe)

Ange Bigeni, The universal sl2 weight system and the Kreweras triangle, arXiv:1712.05475 [math.CO], 2017.

Ange Bigeni, Combinatorial interpretations of the Kreweras triangle in terms of subset tuples, arXiv:1712.01929 [math.CO], 2017.

Ange Bigeni, A generalization of the Kreweras triangle through the universal sl_2 weight system, Journal of Combinatorial Theory, Series A (2019) Vol. 161, 309-326.

Alexander Burstein, Sergi Elizalde and Toufik Mansour, Restricted Dumont permutations, Dyck paths and noncrossing partitions, arXiv:math/0610234 [math.CO], 2006. [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.

Richard Ehrenborg & Einar Steingrímsson, Yet another triangle for the Genocchi numbers, European J. Combin. 21 (2000), no. 5, 593-600. MR1771988 (2001h:05008).

I. M. Gessel, Applications of the classical umbral calculus, arXiv:math/0108121 [math.CO], 2001.

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.

Gábor Hetyei, Alternation acyclic tournaments, arXiv:math/1704.07245 [math.CO], 2017.

G. Kreweras, Sur les permutations comptées par les nombres de Genocchi de 1-ière et 2-ième espèce, Europ. J. Comb., vol. 18, pp. 49-58, 1997. (See also page 76.)

Alexander Lazar and Michelle L. Wachs, The Homogenized Linial Arrangement and Genocchi Numbers, arXiv:1910.07651 [math.CO], 2019.

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

L. Seidel, Über eine einfache Entstehungsweise der Bernoulli'schen Zahlen und einiger verwandten Reihen, Sitzungsberichte der mathematisch-physikalischen Classe der königlich bayerischen Akademie der Wissenschaften zu München, volume 7 (1877), 157-187.

G. Viennot, Interprétations combinatoires des nombres d'Euler et de Genocchi, Seminar on Number Theory, 1981/1982, Exp. No. 11, 94 pp., Univ. Bordeaux I, Talence, 1982.

FORMULA

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

a(n) = A000366(n)*2^(n-1).

a(n) = 2 * (-1)^n * Sum_{k=0..n} binomial(n, k)*(1-2^(n+k+1))*B(n+k+1), with B(n) the Bernoulli numbers. - Ralf 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, Oct 07 2005

G.f.: (of 1,1,2,8,...) 1/(1-x-x^2/(1-5*x-16*x^2/(1-13*x-81*x^2/(1-25*x-256*x^2/(1-41*x-625*x^2/(1-... (continued fraction). - Paul Barry, Nov 27 2009

O.g.f.: Sum_{n>=0} n!*(n+1)! * x^(n+1) / Product_{k=1..n} (1 + k*(k+1)*x). - Paul D. Hanna, May 10 2012

From Sergei N. Gladkovskii, Dec 14 2011, Dec 27 2012, May 29 2013, Oct 09 2013, Oct 24 2013, Oct 27 2013: (Start) Continued fractions:

G.f.: A(x) = 1/S(0), S(k) = 1 - x*(k+1)*(k+2)/(1 - x*(k+1)*(k+2)/S(k+1)).

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).

G.f.: A(x) = (1/(G(0)-1)/x where G(k) = 1 - x*(k+1)^2/(1 - x*(k+1)^2/G(k+1)).

G.f.: 2/G(0), where G(k) = 1 + 1/(1 - 1/(1 - 1/(4*x*(k+1)) + 1/G(k+1))).

G.f.: Q(0)/x - 1/x, where Q(k) = 1 - x*(k+1)^2/( x*(k+1)^2 - 1/(1 - x*(k+1)^2/( x*(k+1)^2 - 1/Q(k+1)))).

G.f.: T(0)/(1-2*x), where T(k) = 1 - x^2*((k + 2)*(k+1))^2/(x^2*((k + 2)*(k+1))^2 - (1 - 2*x*k^2 - 4*x*k - 2*x)*(1 - 2*x*k^2 - 8*x*k - 8*x)/T(k+1)).

G.f.: R(0), where R(k) = 1 - x*(k+1)*(k+2)/( x*(k+1)*(k+2) - 1/(1 - x*(k+1)*(k+2)/( x*(k+1)*(k+2) - 1/R(k+1) ))). (End)

a(n) ~ 2^(2*n+4) * n^(2*n+3/2) / (exp(2*n) * Pi^(2*n+1/2)). - Vaclav Kotesovec, Oct 28 2014

MAPLE

seq(2*(-1)^n*add(binomial(n, k)*(1 - 2^(n+k+1))*bernoulli(n+k+1), k=0..n), n=1..20); # G. C. Greubel, Oct 18 2019

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}] (* 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])) \\ Paul D. Hanna

(Sage) # Algorithm of L. Seidel (1877)

# n -> [a(1), ..., a(n)] for n >= 1.

def A005439_list(n) :

    D = []; [D.append(0) for i in (0..n+2)]; D[1] = 1

    R = [] ; b = True

    for i in(0..2*n-1) :

        h = i//2 + 1

        if b :

            for k in range(h-1, 0, -1) : D[k] += D[k+1]

        else :

            for k in range(1, h+1, 1) :  D[k] += D[k-1]

        if b : R.append(D[1])

        b = not b

    return R

A005439_list(18) # Peter Luschny, Apr 01 2012

(Sage) [2*(-1)^n*sum(binomial(n, k)*(1-2^(n+k+1))*bernoulli(n+k+1) for k in (0..n)) for n in (1..20)] # G. C. Greubel, Oct 18 2019

(MAGMA) [2*(-1)^n*(&+[Binomial(n, k)*(1-2^(n+k+1))*Bernoulli(n+k+1): k in [0..n]]): n in [1..20]]; // G. C. Greubel, Nov 28 2018

(GAP) List([1..20], n->2*(-1)^n*Sum([0..n], k->Binomial(n, k)*(1-2^(n+k+1))*Bernoulli(n+k+1))); # Muniru A Asiru, Nov 29 2018

CROSSREFS

See A000366 for further information.

Sequence in context: A325290 A197949 A243953 * A128814 A108208 A203199

Adjacent sequences:  A005436 A005437 A005438 * A005440 A005441 A005442

KEYWORD

nonn,nice,easy

AUTHOR

Simon Plouffe

EXTENSIONS

More terms and additional comments from David W. Wilson, Jan 11 2001

STATUS

approved

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Last modified November 14 12:21 EST 2019. Contains 329114 sequences. (Running on oeis4.)