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A000657 Median Euler numbers (the middle numbers of Arnold's shuttle triangle). 9
1, 1, 4, 46, 1024, 36976, 1965664, 144361456, 13997185024, 1731678144256, 266182076161024, 49763143319190016, 11118629668610842624, 2925890822304510631936, 895658946905031792553984, 315558279782214450517374976 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

Also central terms of the triangle in A008280. - Reinhard Zumkeller, Nov 01 2013

REFERENCES

V. I. Arnold, The calculus of snakes and the combinatorics of Bernoulli, Euler and Springer numbers of Coxeter groups, Uspekhi Mat. nauk., 47 (#1, 1992), 3-45 = Russian Math. Surveys, Vol. 47 (1992), 1-51.

V. I. Arnold, Springer numbers and Morsification spaces. J. Algebraic Geom. 1 (1992), no. 2, 197-214.

D. Dumont, Further triangles of Seidel-Arnold type and continued fractions related to Euler and Springer numbers, Adv. Appl. Math., 16 (1995), 275-296.

L. Seidel, Über eine einfache Entstehungsweise der Bernoulli'schen Zahlen und einiger verwandten Reihen, Sitzungsberichte der mathematisch-physikalischen Classe der königlich bayerischen Akademie der Wissenschaften zu München, volume 7 (1877), 157-187.

LINKS

Vincenzo Librandi, Table of n, a(n) for n = 0..200

A. Randrianarivony and J. Zeng, Une famille de polynomes qui interpole plusieurs suites..., Adv. Appl. Math. 17 (1996), 1-26.

FORMULA

Row sums of triangle, read by rows, [0, 1, 4, 9, 16, 25, 36, 49, ...] DELTA [1, 2, 6, 5, 11, 8, 16, 11, 21, 14, ...] where DELTA is Deléham's operator defined in A084938.

G.f.: Sum[n>=0, a(n)x^n] = 1/(1-1*1x/(1-1*3x/(1-2*5x/(1-2*7x/(1-3*9x/...))))). - Ralf Stephan, Sep 09 2004

G.f.: 1/G(0) where G(k) =  1 - x*(8*k^2+4*k+1) - x^2*(k+1)^2*(4*k+1)*(4*k+3)/G(k+1);  (recursively defined continued fraction). - Sergei N. Gladkovskii, Feb 05 2013

G.f.: G(0)/(1-x), where G(k) = 1 - x^2*(k+1)^2*(4*k+1)*(4*k+3)/( x^2*(k+1)^2*(4*k+1)*(4*k+3) - (1 - x*(8*k^2+4*k+1))*(1 - x*(8*k^2+20*k+13))/G(k+1) ); (continued fraction). - Sergei N. Gladkovskii, Feb 01 2014

MAPLE

Digits := 40: rr := array(1..40, 1..40): rr[1, 1] := 1: for i from 1 to 39 do rr[i+1, 1] := subs(x=0, diff(1+tan(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]-(-1)^i*rr[i-1, j-1]: od: od: [seq(rr[2*i-1, i], i=1..20)];

MATHEMATICA

max = 20; rr[1, 1] = 1; For[i = 1, i <= 2*max - 1, i++, rr[i + 1, 1] = D[1 + Tan[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] - (-1)^i*rr[i - 1, j - 1]]]; Table[rr[2*i - 1, i], {i, 1, max}] (* Jean-François Alcover, Jul 10 2012, after Maple *)

PROG

(Sage) Algorithm of L. Seidel (1877)

def A000657_list(n) :

    R = []; A = {-1:0, 0:1}

    k = 0; e = 1

    for i in (0..n) :

        Am = 0; A[k + e] = 0; e = -e

        for j in (0..i) :

            Am += A[k]; A[k] = Am; k += e

        if e < 0 :

            R.append(A[0])

    return R

A000657_list(30)  # Peter Luschny, Apr 02 2012

(Haskell)

a000657 n = a008280 (2 * n) n  -- Reinhard Zumkeller, Nov 01 2013

CROSSREFS

Cf. A084938, A002832. For a signed version see A099023.

Related polynomials in A098277.

Sequence in context: A126739 A191870 A099023 * A001623 A188634 A210855

Adjacent sequences:  A000654 A000655 A000656 * A000658 A000659 A000660

KEYWORD

nonn,nice,easy

AUTHOR

N. J. A. Sloane, Don Knuth

EXTENSIONS

More terms from Barbara Haas Margolius (margolius(AT)math.csuohio.edu) Feb 12 2001, corrected by S. A. Irvine, Dec 22 2010

STATUS

approved

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Last modified November 23 16:15 EST 2014. Contains 249851 sequences.