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%I #39 Sep 29 2023 14:24:57
%S 1,2,28,1112,87568,11447072,2239273408,612359887232,223061763490048,
%T 104399900177326592,61049165415292607488,43617245341775265585152,
%U 37385513306142843500105728,37862584188750782065354022912
%N Counts a family of permutations occurring in the study of squeezed states of the simple harmonic oscillator.
%C The sequence a(n), with the convention a(0) = 1, enumerates permutations p(1)p(2)...p(4*n) in the symmetric group on 4*n letters having the following properties:
%C 1) The permutation can be written as a product of disjoint two cycles.
%C 2) For i = 1,...,2*n, positions 2*i-1 and 2*i are either both ascents (labeled A) or both descents (labeled D).
%C The set of permutations satisfying condition (1) forms a subgroup of Symm(4*n) of order A001147(4*n).
%C Here are some examples of permutations (written in cycle form) in Symm(8), satisfying these conditions, together with their ascent-descent labelings.
%C ... (14)(23)(57)(68) of type AADDAADD;
%C ... (15)(26)(37)(48) of type AAAADDDD.
%C Since the permutations being considered consist of disjoint 2-cycles their ascent-descent labelings must have an equal number of A's and D's.
%C Further examples can be found in the Example section below.
%C This family of permutations have arisen in the study of squeezed states
%C of the simple harmonic oscillator [Sukumar and Hodges].
%C See A186492 for a recursive triangle to compute this sequence.
%H Jitender Singh, <a href="http://arxiv.org/abs/1402.0065">On an arithmetic convolution</a>, arXiv:1402.0065 [math.NT], 2014 and <a href="https://cs.uwaterloo.ca/journals/JIS/VOL17/Singh/singh8.html">J. Int. Seq. 17 (2014) # 14.6.7</a>.
%H C. V. Sukumar and A. Hodges, <a href="https://doi.org/10.1098/rspa.2007.0003">Quantum algebras and parity-dependent spectra</a>, Proc. R. Soc. A (2007) 463.
%F GENERATING FUNCTION
%F (1)... sqrt(sec(2*x)) = Sum_{n>=0} a(n)*x^(2*n)/(2*n)!
%F = 1 + 2*x^2/2! + 28*x^4/4! + 1112*x^6/6! + ....
%F Compare with the e.g.f. Of A000364.
%F O.g.f. as a continued fraction: 1/(1-2*x/(1-12*x/(1-30*x/(...-2*n*(2*n-1)*x/(1-...))))) = 1 + 2*x + 28*x^2 + 1112*x^3 + ....
%F From _Sergei N. Gladkovskii_, Oct 23 2012: (Start)
%F G.f.: 1/U(0) where U(k) = 1 - (4*k+1)*(4*k+2)*x/( 1 - (4*k+3)*(4*k+4)*x/U(k+1)); (continued fraction, 2-step).
%F G.f.: 1/S(0) where S(k) = 1 - 2*x*(16*k^2 + 4*k + 1) - 8*x^2*(k+1)*(2*k+1)*(4*k+1)*(4*k+3)/S(k+1); (continued fraction, 1-step).
%F (End)
%F Let A(x) = Sum_{n>=0} a(n)*x^n = 1/T(0) where T(k)= 1 - (2*k+1)*(2*k+2)*x^2/T(k+1) -(continued fraction, 1-step),- then sqrt(sec(2*x)) = Sum_{n>=0} a(n)*x^n/n!. - _Sergei N. Gladkovskii_, Oct 25 2012
%F G.f.: 1/S(0) where S(k)= 1 - (2*k+1)*(2*k+2)*x /S(k+1); (continued fraction, 1-step). - _Sergei N. Gladkovskii_, Oct 26 2012
%F G.f.: Q(0), where Q(k) = 1 - x*(2*k+1)*(2*k+2)/(x*(2*k+1)*(2*k+2) - 1/Q(k+1)); (continued fraction). - _Sergei N. Gladkovskii_, Oct 09 2013
%F For n > 0, a(n) = Sum_{k=1..n} a(n-k)*binomial(2*n,2*k)*(k/(2*n)-1)*(-4)^k. - _Tani Akinari_, Sep 19 2023.
%e a(1)=2:
%e The two permutations in Symm(4) satisfying the conditions are
%e ... (13)(24) of type AADD
%e ... (14)(23) of type AADD.
%e a(2)=28:
%e Clearly, the ascent-descent structure of one of our permutations must start with an AA and finish with a DD so the two possible types are AAAADDDD and AADDAADD.
%e There are 4!=24 permutations of type AAAADDDD coming from the bijections of {1,2,3,4} onto {5,6,7,8}.
%e There are 2*2 = 4 permutations of the remaining type AADDAADD, namely
%e ... (13)(24)(57)(68)
%e ... (13)(24)(58)(67)
%e ... (14)(23)(57)(68)
%e ... (14)(23)(58)(67).
%p G:= sqrt(sec(2*x)): Gser := series(G, x = 0,32):
%p seq((2*n)!*coeff(Gser,x^(2*n)), n = 1..15);
%p # Alternative, using the Singh transformation 'g' from Maple in A126156:
%p a := n -> (-4)^n*g(euler, 2*n);
%p seq(a(n), n = 0..13); # _Peter Luschny_, Sep 29 2023
%o (Maxima) a[n]:=if n=0 then 1 else sum(a[n-k]*binomial(2*n,2*k)*(k/(2*n)-1)*(-4)^k,k,1,n);
%o makelist(a[n],n,0,20); /* _Tani Akinari_, Sep 19 2023 */
%Y Cf. A000364, A186492, A126156.
%K nonn,easy
%O 0,2
%A _Peter Bala_, Feb 22 2011
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