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A000657 Median Euler numbers (the middle numbers of Arnold's shuttle triangle).

%I A000657 #94 Nov 24 2024 17:11:17
%S A000657 1,1,4,46,1024,36976,1965664,144361456,13997185024,1731678144256,
%T A000657 266182076161024,49763143319190016,11118629668610842624,
%U A000657 2925890822304510631936,895658946905031792553984,315558279782214450517374976,126780706777739389745128013824
%N A000657 Median Euler numbers (the middle numbers of Arnold's shuttle triangle).
%C A000657 Also central terms of the triangle in A008280. - _Reinhard Zumkeller_, Nov 01 2013
%C A000657 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
%D A000657 V. I. Arnold, Springer numbers and Morsification spaces. J. Algebraic Geom. 1 (1992), no. 2, 197-214.
%D A000657 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.
%H A000657 Vincenzo Librandi, <a href="/A000657/b000657.txt">Table of n, a(n) for n = 0..200</a>
%H A000657 V. I. Arnold, <a href="http://mi.mathnet.ru/eng/umn4470">The calculus of snakes and the combinatorics of Bernoulli, Euler and Springer numbers of Coxeter groups</a>, Uspekhi Mat. nauk., 47 (#1, 1992), 3-45 = Russian Math. Surveys, Vol. 47 (1992), 1-51.
%H A000657 Ange Bigeni and Evgeny Feigin, <a href="https://arxiv.org/abs/1808.04275">Symmetric Dellac configurations</a>, arXiv:1808.04275 [math.CO], 2018.
%H A000657 D. Dumont, <a href="http://dx.doi.org/10.1006/aama.1995.1014">Further triangles of Seidel-Arnold type and continued fractions related to Euler and Springer numbers</a>, Adv. Appl. Math., 16 (1995), 275-296.
%H A000657 A. Randrianarivony and J. Zeng, <a href="http://dx.doi.org/10.1006/aama.1996.0001">Une famille de polynomes qui interpole plusieurs suites...</a>, Adv. Appl. Math. 17 (1996), 1-26.
%F A000657 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.
%F A000657 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
%F A000657 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
%F A000657 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
%F A000657 a(n) = (-1)^(n)*Sum_{k=0..n} C(n,k)*Euler(n+k). - _Vladimir Kruchinin_, Apr 06 2015
%F A000657 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
%F A000657 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
%p A000657 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)];
%p A000657 # Alternatively after _Alois P. Heinz_ in A000111:
%p A000657 b := proc(u, o) option remember;
%p A000657 `if`(u + o = 0, 1, add(b(o - 1 + j, u - j), j = 1..u)) end:
%p A000657 a := n -> b(n, n): seq(a(n), n = 0..15); # _Peter Luschny_, Oct 27 2017
%t A000657 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 A000657 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_ *)
%o A000657 (Sage) # Algorithm of L. Seidel (1877)
%o A000657 def A000657_list(n) :
%o A000657     R = []; A = {-1:0, 0:1}
%o A000657     k = 0; e = 1
%o A000657     for i in (0..n) :
%o A000657         Am = 0; A[k + e] = 0; e = -e
%o A000657         for j in (0..i) :
%o A000657             Am += A[k]; A[k] = Am; k += e
%o A000657         if e < 0 :
%o A000657             R.append(A[0])
%o A000657     return R
%o A000657 A000657_list(30)  # Peter Luschny, Apr 02 2012
%o A000657 (Haskell)
%o A000657 a000657 n = a008280 (2 * n) n  -- _Reinhard Zumkeller_, Nov 01 2013
%o A000657 (Maxima)
%o A000657 a(n):=(-1)^(n)*sum(binomial(n,k)*euler(n+k),k,0,n); /* _Vladimir Kruchinin_, Apr 06 2015 */
%Y A000657 Cf. A084938, A002832. For a signed version see A099023.
%Y A000657 Related polynomials in A098277.
%Y A000657 A diagonal of A323834.
%Y A000657 Cf. A005799.
%K A000657 nonn,nice,easy
%O A000657 0,3
%A A000657 _N. J. A. Sloane_, _Don Knuth_
%E A000657 More terms from Barbara Haas Margolius (margolius(AT)math.csuohio.edu), Feb 12 2001
%E A000657 Corrected by _Sean A. Irvine_, Dec 22 2010