%I M1979 #73 Sep 28 2024 01:44:52

%S 1,1,2,10,104,1816,47312,1714000,82285184,5052370816,386051862272,

%T 35917232669440,3996998043812864,524203898507631616,

%U 80011968856686405632,14061403972845412526080,2818858067801804443910144

%N Generalized Euler numbers of type 2^n.

%C Also, a(n) equals the number of alternating permutations (p(1),...,p(2n)) of the multiset {1,1,2,2,...,n,n} satisfying p(1) <= p(2) > p(3) <= p(4) > p(5) <= ... <= p(2n). Hence, A275801(n) <= a(n) <= A275829(n). - _Max Alekseyev_, Aug 10 2016

%C This is the BinomialMean transform of A000364 (see A075271 for definition of transform). - _John W. Layman_, Dec 04 2002

%C This sequence appears to be middle column in Poupard's triangle A008301.

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

%H Peter Bala, <a href="/A002439/a002439.pdf">Some S-fractions related to the expansions of sin(ax)/cos(bx) and cos(ax)/cos(bx)</a>

%H Shishuo Fu, Zhicong Lin, and Zhi-Wei Sun, <a href="https://arxiv.org/abs/2109.11506">Proof of several conjectures relating permanents to Combinatorial sequences</a>, arXiv:2109.11506v3 [math.CO], 2021-2023.

%H Shishuo Fu, Zhicong Lin, and Zhi-Wei Sun, <a href="https://doi.org/10.1016/j.aam.2024.102789">Permanent identities, combinatorial sequences, and permutation statistics</a>, Advances in Applied Mathematics, Volume 163, Part A, 102789 (2025).

%H Ira M. Gessel, <a href="http://dx.doi.org/10.1016/0097-3165(90)90060-A">Symmetric functions and P-recursiveness</a>, J. Combin. Theory Ser. A 53 (1990), no. 2, 257-285.

%H H. Prodinger, <a href="https://arxiv.org/abs/1102.5186">On Touchard's continued fraction and extensions: combinatorics-free, self-contained proofs </a>, arXiv:1102.5186 [math.CO], 2011.

%H Michael Z. Spivey and Laura L. Steil, <a href="http://www.cs.uwaterloo.ca/journals/JIS/VOL9/Spivey/spivey7.html">The k-Binomial Transforms and the Hankel Transform</a>, Journal of Integer Sequences, Vol. 9 (2006), Article 06.1.1.

%F a(n) = (1/2^n) * Sum_{i=0..n} binomial(n, i) * A000364(i).

%F From _Sergei N. Gladkovskii_, Dec 27 2012, Oct 11 2013, Oct 27 2013, Jan 08 2014: (Start) Continued fractions:

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

%F G.f.: Q(0)/(1-x), where Q(k) = 1 - x^2*(k+1)^2*(2*k+1)^2/(x^2*(k+1)^2*(2*k+1)^2 - (4*x*k^2 + 2*x*k + x - 1)*( 4*x*k^2 + 10*x*k + 7*x - 1)/Q(k+1)).

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

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

%F a(n) ~ 2^(3*n+3) * n^(2*n+1/2) / (exp(2*n) * Pi^(2*n+1/2)). - _Vaclav Kotesovec_, May 30 2015

%F a(n) = 2^n * Sum_{k=0..n} (-1)^k*binomial(n, k)*euler(n+k, 1). - _Peter Luschny_, Aug 23 2017

%F From _Peter Bala_, Dec 21 2019: (Start)

%F O.g.f. as a continued fraction: 1/(1 - x/(1 - x/(1 - 6*x/(1 - 6*x/(1 - 15*x/(1 - 15*x/(1 - ... - n*(2*n-1)*x/(1 - n*(2*n-1)*x/(1 - ...))))))))) - apply Bala, Proposition 3, with a = 0, b = 1 and replace x with x/2.

%F Conjectures:

%F E.g.f. as a continued fraction: 2/(2 - (1-exp(-4*t))/(2 - (1-exp(-8*t))/(2 - (1-exp(-12*t))/(2 - ... )))) = 1 + t + 2*t^2/2! + 10*t^3/3! + 104*t^4/4! + ....

%F Cf. A000657. [added April 18 2024: for a proof of this conjecture see Fu et al., Section 4.3.]

%F a(n) = (-2)^(n+1)*Sum_{k = 0..floor((n-1)/2)} binomial(n,2*k+1)*(2^(2*n-2*k) - 1)*Bernoulli(2*n-2*k)/(2*n-2*k) for n >= 1. (End)

%p T := proc(n, k) option remember;

%p if n < 0 or k < 0 then 0

%p elif n = 0 then euler(k, 1)

%p else T(n-1, k+1) - T(n-1, k) fi end:

%p a := n -> (-2)^n*T(n, n); seq(a(n), n=0..16); # _Peter Luschny_, Aug 23 2017

%t a[n_] := Sum[Binomial[n, i]Abs[EulerE[2i]], {i, 0, n}]/2^n

%Y Right edge of triangle A210108.

%Y Cf. A000364, A008301, A275801, A275829, A154283, A000657.

%K nonn,easy

%O 0,3

%A _N. J. A. Sloane_, _Simon Plouffe_

%E Edited by _Dean Hickerson_, Dec 10 2002