cp's OEIS Frontend

A117513 Number of ways of arranging 2*n tokens in a row, with 2 copies of each token from 1 through n, such that between every pair of tokens labeled i (i = 1..n-1) there is exactly one taken labeled i+1.

%I A117513 #23 Jul 26 2021 10:58:57
%S A117513 1,2,12,136,2480,66336,2446528,118984832,7378078464,568142287360,
%T A117513 53189920492544,5949749335001088,783686338494312448,
%U A117513 120058889459865165824,21166245289132322242560,4254864627502524070395904,967406173145278971994898432,247007221085479721384365129728
%N A117513 Number of ways of arranging 2*n tokens in a row, with 2 copies of each token from 1 through n, such that between every pair of tokens labeled i (i = 1..n-1) there is exactly one taken labeled i+1.
%C A117513 From _Paul Barry_, Oct 12 2009: (Start)
%C A117513 The aerated sequence is (2^(n/2 - 1) + 0^(n/2)/2)*((1 + (-1)^n)/2)*n!*[x^n](1 + x*tan(x/2)).
%C A117513 Multiples of the unsigned Genocchi numbers A110501: (1, 1, 3, 17, 155,...)*(1, 2, 4, 8, 16,...). (End)
%H A117513 Michael De Vlieger, <a href="/A117513/b117513.txt">Table of n, a(n) for n = 1..258</a>
%H A117513 Guo-Niu Han, <a href="https://arxiv.org/abs/2006.14070">Enumeration of Standard Puzzles</a>, arXiv:2006.14070 [math.CO], 2020.
%H A117513 Guo-Niu Han, <a href="/A196265/a196265.pdf">Enumeration of Standard Puzzles</a> [Cached copy]
%F A117513 G.f.: 1/(1-2*x/(1-4*x/(1-8*x/(1-12*x/(1-18*x/(1-24*x/(1-32*x/(1-.../(1-2* floor((n+2)^2/4)*x/(1-... (continued fraction). - _Paul Barry_, Dec 03 2009
%F A117513 G.f.: T(0), where T(k) = 1 - x*(2*k+2)*(k+1)/( x*(2*k+2)*(k+1) - 1/( 1 - x*(2*k+2)*(k+2)/( x*(2*k+2)*(k+2) - 1/T(k+1) ))); (continued fraction). - _Sergei N. Gladkovskii_, Oct 24 2013
%F A117513 a(n) = (-2)^n*(1 - 2^(2*n))*Bernoulli(2*n). - _Peter Luschny_, Jul 26 2021
%p A117513 a := n -> (-2)^n*(1 - 2^(2*n))*bernoulli(2*n);
%p A117513 seq(a(n), n = 1..18); # _Peter Luschny_, Jul 26 2021
%t A117513 Array[(-2)^#*(1 - 2^(2 #))*BernoulliB[2 #] &, 18] (* _Michael De Vlieger_, Jul 26 2021 *)
%o A117513 (Sage) # Algorithm of L. Seidel (1877)
%o A117513 # n -> [a(1), ..., a(n)] for n >= 1.
%o A117513 def A117513_list(n) :
%o A117513     D = [0]*(n+2); D[1] = 1
%o A117513     R = []; z = 1/2; b = True
%o A117513     for i in(0..2*n-1) :
%o A117513         h = i//2 + 1
%o A117513         if b :
%o A117513             for k in range(h-1, 0, -1) : D[k] += D[k+1]
%o A117513             z *= 2
%o A117513         else :
%o A117513             for k in range(1, h+1, 1) :  D[k] += D[k-1]
%o A117513         b = not b
%o A117513         if b : R.append(D[h]*z)
%o A117513     return R
%o A117513 A117513_list(15) # _Peter Luschny_, Jun 29 2012
%Y A117513 Cf. A002105, A110501, A117514, A117515.
%K A117513 nonn
%O A117513 1,2
%A A117513 Nan Zang (nzang(AT)cs.ucsd.edu), Apr 28 2006
%E A117513 More terms from _Paul Barry_, Oct 12 2009