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%I #70 Aug 02 2021 14:37:48
%S 1,3,24,402,11616,514608,32394624,2748340752,302234850816,
%T 41811782731008,7106160248346624,1455425220196234752,
%U 353536812021243273216,100492698847094242603008,33045185784774350171111424
%N Median Euler numbers.
%C There are two kinds of Euler median numbers, the 'right' median numbers (this sequence), and the 'left' median numbers (A000657).
%C Apparently all terms (except the initial 1) have 3-valuation 1. - _F. Chapoton_, Aug 02 2021
%H Vincenzo Librandi, <a href="/A002832/b002832.txt">Table of n, a(n) for n = 1..100</a>
%H Ange Bigeni and Evgeny Feigin, <a href="https://arxiv.org/abs/1808.04275">Symmetric Dellac configurations</a>, arXiv:1808.04275 [math.CO], 2018.
%H Kwang-Wu Chen, <a href="http://www.cs.uwaterloo.ca/journals/JIS/VOL6/Chen/chen50.html">An Interesting Lemma for Regular C-fractions</a>, J. Integer Seqs., Vol. 6, 2003.
%H D. Dumont, <a href="http://dx.doi.org/10.1006/aama.1995.1014">Further triangles of Seidel-Arnold type and continued fractions related to Euler and Springer numbers</a>, Adv. Appl. Math., 16 (1995), 275-296.
%H A. Randrianarivony and J. Zeng, <a href="http://dx.doi.org/10.1006/aama.1996.0001">Une famille de polynômes qui interpole plusieurs suites...</a>, Adv. Appl. Math. 17 (1996), 1-26. (In French, with a summary in English on p. 1).
%H R. C. Read, <a href="/A002832/a002832.pdf">Letter to N. J. A. Sloane, 1992</a>
%F G.f.: Sum_{n>=0} a(n)*x^n = 1/(1-1*3x/(1-1*5x/(1-2*7x/(1-2*9x/(1-3*11x/...))))).
%F G.f.: -1/G(0) where G(k)= x*(8*k^2+8*k+3) - 1 - (4*k+5)*(4*k+3)*(k+1)^2*x^2/G(k+1); (continued fraction, 1-step). - _Sergei N. Gladkovskii_, Aug 08 2012
%F a(n) ~ 2^(4*n+3/2) * n^(2*n-1/2) / (exp(2*n) * Pi^(2*n-1/2)). - _Vaclav Kotesovec_, Apr 23 2015
%p rr := array(1..40,1..40):rr[1,1] := 0:for i from 1 to 39 do rr[i+1,1] := (subs(x=0,diff((exp(x)-1)/cosh(x),x$i))):od: for i from 2 to 40 do for j from 2 to i do rr[i,j] := rr[i,j-1]-rr[i-1,j-1]:od:od: seq(rr[2*i-1,i-1],i=2..20); # Barbara Haas Margolius (margolius(AT)math.csuohio.edu) Feb 16 2001, corrected by _R. J. Mathar_, Dec 22 2010
%t max = 20; rr[1, 1] = 0; For[i = 1, i <= 2*max - 1, i++, rr[i + 1, 1] = D[(Exp[x] - 1)/Cosh[x], {x, i}] /. x -> 0]; For[i = 2, i <= 2*max, i++, For[j = 2, j <= i, j++, rr[i, j] = rr[i, j - 1] - rr[i - 1, j - 1]]]; Table[(-1)^i*rr[2*i - 1, i - 1], {i, 2, max}] (* _Jean-François Alcover_, Jul 10 2012, after Maple *)
%Y Cf. A000657.
%Y See related polynomials in A098277.
%Y A diagonal of A323833.
%K nonn
%O 1,2
%A _N. J. A. Sloane_, Dec 11 1996
%E More terms from Barbara Haas Margolius (margolius(AT)math.csuohio.edu), Feb 16 2001
%E Terms corrected by _R. J. Mathar_, Dec 22 2010
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