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A002832
Median Euler numbers.
7
1, 3, 24, 402, 11616, 514608, 32394624, 2748340752, 302234850816, 41811782731008, 7106160248346624, 1455425220196234752, 353536812021243273216, 100492698847094242603008, 33045185784774350171111424
OFFSET
1,2
COMMENTS
There are two kinds of Euler median numbers, the 'right' median numbers (this sequence), and the 'left' median numbers (A000657).
Apparently all terms (except the initial 1) have 3-valuation 1. - F. Chapoton, Aug 02 2021
LINKS
Ange Bigeni and Evgeny Feigin, Symmetric Dellac configurations, arXiv:1808.04275 [math.CO], 2018.
Kwang-Wu Chen, An Interesting Lemma for Regular C-fractions, J. Integer Seqs., Vol. 6, 2003.
A. Randrianarivony and J. Zeng, Une famille de polynômes qui interpole plusieurs suites..., Adv. Appl. Math. 17 (1996), 1-26. (In French, with a summary in English on p. 1).
FORMULA
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/...))))).
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
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
MAPLE
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
MATHEMATICA
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 *)
CROSSREFS
Cf. A000657.
See related polynomials in A098277.
A diagonal of A323833.
Sequence in context: A193210 A065761 A374021 * A233151 A236466 A371126
KEYWORD
nonn
AUTHOR
N. J. A. Sloane, Dec 11 1996
EXTENSIONS
More terms from Barbara Haas Margolius (margolius(AT)math.csuohio.edu), Feb 16 2001
Terms corrected by R. J. Mathar, Dec 22 2010
STATUS
approved