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A000657
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Median Euler numbers (the middle numbers of Arnold's shuttle triangle).
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12
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1, 1, 4, 46, 1024, 36976, 1965664, 144361456, 13997185024, 1731678144256, 266182076161024, 49763143319190016, 11118629668610842624, 2925890822304510631936, 895658946905031792553984, 315558279782214450517374976, 126780706777739389745128013824
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OFFSET
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0,3
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COMMENTS
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Conjecture: taking the sequence modulo an integer k gives an eventually purely periodic sequence with period dividing phi(k). For example, the sequence taken modulo 9 begins [1, 1, 4, 1, 7, 4, 1, 7, 4, 1, 7, ...] with an apparent period [4, 1, 7] of length 3 = phi(9)/2 beginning at a(2). - Peter Bala, May 08 2023
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REFERENCES
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V. I. Arnold, Springer numbers and Morsification spaces. J. Algebraic Geom. 1 (1992), no. 2, 197-214.
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.
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LINKS
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FORMULA
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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
a(n) ~ 2^(4*n+5/2) * n^(2*n+1/2) / (exp(2*n) * Pi^(2*n+1/2)). - Vaclav Kotesovec, Apr 06 2015
Conjectural e.g.f. as a continued fraction: 1/(1 - (1 - exp(-2*t))/(2 - (1 - exp(-4*t))/(1 - (1 - exp(-6*t))/(2 - (1 - exp(-8*t))/(1 - ... )))) = 1 + t + 4*t^2/2! + 46*t^3/3! + .... Cf. A005799. - Peter Bala, Dec 26 2019
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MAPLE
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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)];
b := proc(u, o) option remember;
`if`(u + o = 0, 1, add(b(o - 1 + j, u - j), j = 1..u)) end:
a := n -> b(n, n): seq(a(n), n = 0..15); # Peter Luschny, Oct 27 2017
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MATHEMATICA
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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 *)
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PROG
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(Sage) # Algorithm of L. Seidel (1877)
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)
(Maxima)
a(n):=(-1)^(n)*sum(binomial(n, k)*euler(n+k), k, 0, n); /* Vladimir Kruchinin, Apr 06 2015 */
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CROSSREFS
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KEYWORD
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nonn,nice,easy
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AUTHOR
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EXTENSIONS
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More terms from Barbara Haas Margolius (margolius(AT)math.csuohio.edu), Feb 12 2001
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STATUS
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approved
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