The OEIS is supported by the many generous donors to the OEIS Foundation.

 Year-end appeal: Please make a donation to the OEIS Foundation to support ongoing development and maintenance of the OEIS. We are now in our 60th year, we have over 367,000 sequences, and we’ve reached 11,000 citations (which often say “discovered thanks to the OEIS”). Other ways to Give
 Hints (Greetings from The On-Line Encyclopedia of Integer Sequences!)
 A000657 Median Euler numbers (the middle numbers of Arnold's shuttle triangle). 12
 1, 1, 4, 46, 1024, 36976, 1965664, 144361456, 13997185024, 1731678144256, 266182076161024, 49763143319190016, 11118629668610842624, 2925890822304510631936, 895658946905031792553984, 315558279782214450517374976, 126780706777739389745128013824 (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 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 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. 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 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 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 (Maxima) a(n):=(-1)^(n)*sum(binomial(n, k)*euler(n+k), k, 0, n); /* Vladimir Kruchinin, Apr 06 2015 */ CROSSREFS Cf. A084938, A002832. For a signed version see A099023. Related polynomials in A098277. A diagonal of A323834. Cf. A005799. Sequence in context: A126739 A191870 A099023 * A356901 A001623 A188634 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 Sean A. Irvine, Dec 22 2010 STATUS approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recents
The OEIS Community | Maintained by The OEIS Foundation Inc.

Last modified December 4 12:24 EST 2023. Contains 367560 sequences. (Running on oeis4.)