A000657 Median Euler numbers (the middle numbers of Arnold's shuttle triangle).
1, 1, 4, 46, 1024, 36976, 1965664, 144361456, 13997185024, 1731678144256, 266182076161024, 49763143319190016, 11118629668610842624, 2925890822304510631936, 895658946905031792553984, 315558279782214450517374976, 126780706777739389745128013824
Offset: 0
References
- 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.
Links
- Vincenzo Librandi, Table of n, a(n) for n = 0..200
- 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.
- Ange Bigeni and Evgeny Feigin, Symmetric Dellac configurations, arXiv:1808.04275 [math.CO], 2018.
- 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 polynomes qui interpole plusieurs suites..., Adv. Appl. Math. 17 (1996), 1-26.
Crossrefs
Programs
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Haskell
a000657 n = a008280 (2 * n) n -- Reinhard Zumkeller, Nov 01 2013
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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)]; # Alternatively after Alois P. Heinz in A000111: 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
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 *) T[n_,0] := KroneckerDelta[n,0]; T[n_,k_] := T[n,k]=T[n,k-1]+T[n-1,n-k]; Table[T[2n,n], {n,0,16}] (* Oliver Seipel, Nov 24 2024, after Peter Luschny *) -
Maxima
a(n):=(-1)^(n)*sum(binomial(n,k)*euler(n+k),k,0,n); /* Vladimir Kruchinin, Apr 06 2015 */
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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
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
a(n) = (-1)^(n)*Sum_{k=0..n} C(n,k)*Euler(n+k). - Vladimir Kruchinin, Apr 06 2015
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
Extensions
More terms from Barbara Haas Margolius (margolius(AT)math.csuohio.edu), Feb 12 2001
Corrected by Sean A. Irvine, Dec 22 2010
Comments