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A005439
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Genocchi medians (or Genocchi numbers of second kind).
(Formerly M1888)
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25
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1, 1, 2, 8, 56, 608, 9440, 198272, 5410688, 186043904, 7867739648, 401293838336, 24290513745920, 1721379917619200, 141174819474169856, 13266093250285568000, 1415974941618255921152, 170361620874699124637696, 22948071824232932086513664, 3439933090471867097102680064
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OFFSET
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0,3
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COMMENTS
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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, pages 185 and 186). - Don Knuth, Jul 13 2007
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
The n-th Genocchi number of the second kind is also the number of collapsed permutations in (2n) letters. A permutation pi of size 2n is said to be collapsed if 1+floor(k/2) <= pi^{-1}(k) <= n + floor(k/2). There are 2 collapsed permutations of size 4, namely 1234 and 1324. - Arvind Ayyer, Oct 23 2020
For any positive integer n, a(n) is (-1)^n times the permanent of the 2n X 2n matrix M with M(j, k) = floor((2*j-k-1)/(2*n)). This former conjecture of Luschny, inspired by a conjecture of Zhi-Wei Sun in A036968, was proven by Fu, Lin and Sun (see link). - Peter Luschny, Sep 07 2021 [updated Sep 24 2021]
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REFERENCES
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N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
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LINKS
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FORMULA
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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) = 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
Rewriting the above: a(n) ~ 4*(2*n+1)! / Pi^(2*n+1). Compare to Genocchi numbers A110501(n) = g_n ~ 4*(2*n)! / Pi^(2*n). So these are indeed like "Genocchi medians" g_{n + 1/2}. - Alan Sokal, May 13 2022
Asymptotic expansion: a(n) ~ 4*(2*n+1)! * Pi^(-(2*n+1)) * (1 + (Pi^2/16)/n + (Pi^2 (Pi^2 - 16)/512)/n^2 + (Pi^2 (Pi^4 + 384)/24576)/n^3 + (Pi^2 (Pi^6 + 96*Pi^4 + 768*Pi^2 - 12288)/1572864)/n^4 + (Pi^2 (Pi^8 + 320*Pi^6 + 12800*Pi^4 + 491520)/125829120)/n^5 + ...) --- Proof uses binomial sum for Genocchi medians in terms of Genocchi or Bernoulli numbers, combined with leading term of convergent sum (with exponentially small corrections) for the latter. Can also check against the 10000 term a-file. - Alan Sokal, May 23 2022.
a(n) = n!^2 * [x^n*y^n] exp(x)*f(x-y), where f(x) is the derivative of the Genocchi number generating function 2*x/(exp(x)+1). - Ira M. Gessel, Jul 23 2024
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MAPLE
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seq(2*(-1)^n*add(binomial(n, k)*(1 - 2^(n+k+1))*bernoulli(n+k+1), k=0..n), n=0..20); # G. C. Greubel, Oct 18 2019
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MATHEMATICA
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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. *)
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PROG
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(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.
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
(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
(Python)
from math import comb
from sympy import bernoulli
def A005439(n): return (-2 if n&1 else 2)*sum(comb(n, k)*(1-(1<<n+k+1))*bernoulli(n+k+1) for k in range(n+1)) # Chai Wah Wu, Apr 14 2023
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CROSSREFS
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KEYWORD
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nonn,nice,easy,changed
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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