A002832 Median Euler numbers.

1, 3, 24, 402, 11616, 514608, 32394624, 2748340752, 302234850816, 41811782731008, 7106160248346624, 1455425220196234752, 353536812021243273216, 100492698847094242603008, 33045185784774350171111424

Offset: 1

Author

N. J. A. Sloane, Dec 11 1996

Keywords

• nonn

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

• Vincenzo Librandi, Table of n, a(n) for n = 1..100
• 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.
• D. Dumont, Further triangles of Seidel-Arnold type and continued fractions related to Euler and Springer numbers, Adv. Appl. Math., 16 (1995), 275-296.
• 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).
• R. C. Read, Letter to N. J. A. Sloane, 1992

Crossrefs

Cf. A000657.

See related polynomials in A098277.

A diagonal of A323833.

Programs

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
# alternative
A002832 := proc(n)
    abs(A323833(n-1,n)) ;
end proc:
seq(A002832(n),n=1..40) ; # R. J. Mathar, Jun 11 2025

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 *)

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

Extensions

More terms from Barbara Haas Margolius (margolius(AT)math.csuohio.edu), Feb 16 2001

Terms corrected by R. J. Mathar, Dec 22 2010