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%I #36 May 26 2021 20:46:03
%S 1,0,0,0,-12,0,0,0,3024,0,0,0,-4390848,0,0,0,21224560896,0,0,0,
%T -257991277243392,0,0,0,6628234834692624384,0,0,0,
%U -319729080846260095008768,0,0,0,26571747463798134334265819136,0,0,0,-3564202847752289659513902717468672,0,0
%N Expansion of the sine lemniscate function sl(x).
%C For the series expansion of the cosine lemniscate cl(x) see A159600. The lemniscatic functions sl(x) and cl(x) played a significant role in the development of mathematics in the 18th and 19th centuries. They were the first examples of elliptic functions. In algebraic number theory all abelian extensions of the Gaussian rationals Q(i) are contained in extensions of Q(i) generated by division values of the lemniscatic functions. - _Peter Bala_, Aug 25 2011
%H Vincenzo Librandi, <a href="/A104203/b104203.txt">Table of n, a(n) for n = 1..229</a>
%H S. Binski, T. R. Hagedorn, <a href="http://www.tcnj.edu/~hagedorn/papers/CapstonePapers/Binski/CapstoneBinskiLemniscate.pdf">Constructions on the Lemniscate</a>
%H D. Dominici, <a href="http://arxiv.org/abs/math/0501052">Nested derivatives: A simple method for computing series expansions of inverse functions</a>, arXiv:math/0501052v2 [math.CA], 2005.
%H A. Gritsans and F. Sadyrbaev, <a href="http://www.lumii.lv/Pages/sbornik1/Lemniscatic-trigonometry-last.pdf">Trigonometry of lemniscatic functions</a>
%H A. Gritsans and F. Sadyrbaev, <a href="http://www.de.dau.lv/matematika/lemniscatic/S3F3v1.pdf">Lemniscatic functions in the theory of the Emden-Fowler differential equation</a>
%H Markus Kuba, Alois Panholzer, <a href="http://arxiv.org/abs/1411.4587">Combinatorial families of multilabelled increasing trees and hook-length formulas</a>, arXiv:1411.4587 [math.CO], 2014.
%H Erik Vigren, Andreas Dieckmann, <a href="https://doi.org/10.3390/sym12061040">Simple Solutions of Lattice Sums for Electric Fields Due to Infinitely Many Parallel Line Charges</a>, Symmetry (2020) Vol. 12, No. 6, 1040.
%H Eric W. Weisstein, <a href="http://mathworld.wolfram.com/LemniscateFunction.html">Lemniscate Function</a>
%F From _Peter Bala_, Aug 25 2011: (Start)
%F The function sl(x) satisfies the differential equation sl''(x) = -2*sl^3(x) with initial conditions sl(0) = 0, sl'(0) = 1.
%F Recurrence relation:
%F a(n+2) = -2*sum {i+j+k = n} n!/(i!*j!*k!)*a(i)*a(j)*a(k).
%F The inverse of the sine lemniscate function may be defined as the algebraic integral
%F sl^(-1)(x) := int {0..x} 1/sqrt(1-s^4)) = x + x^5/10 + x^9/24 + 5*x^13/208 + ....
%F Series reversion produces the expansion
%F sl(x) = x - 12*x^5/5! + 3024*x^9/9! - 4390848*x^13/13! + ....
%F The coefficients in this expansion can be calculated using nested derivatives as follows (see [Dominici, Theorem 4.1): Let f(x) = sqrt(1-x^4). Define the nested derivative D^n[f](x) by means of the recursion
%F D^0[f](x) = 1 and D^(n+1)[f](x) = d/dx(f(x)*D^n[f](x)) for n >= 0.
%F The coefficients in the expansion of D^n[f](x) in powers of f(x) are given in A145271. Then we have a(n) = D^(n-1)[f](0).
%F a(n) is divisible by 12^n and a(n)/12^n produces (a signed and aerated version of) A144853(n).
%F (End)
%F The function sl(x) satisfies the differential equation sl'(x)^2 + sl(x)^4 = 1 with initial conditions sl(0) = 0, sl'(0) = 1. - _Michael Somos_, Oct 12 2019
%e G.f. = x - 12*x^5 + 3024*x^9 - 4390848*x^13 + 21224560896*x^17 + ...
%e Example of the recurrence relation a(n+2) = -2*sum {i+j+k = n} n!/(i!*j!*k!)*a(i)*a(j)*a(k) for n = 13:
%e There are only 6 compositions of 13-2 = 11 that give a nonzero contribution to the sum, namely 11 = 9+1+1 = 1+9+1 = 1+1+9 and 11 = 5+5+1 = 5+1+5 = 1+5+5
%e and hence
%e a(13) = -2*(3*11!/(9!*1!*1*)*a(9)*a(1)*a(1)+3*11!/(5!*5!*1!)*a(5)*a(5)*a(1)) = -4390848.
%t Drop[ Range[0, 37]! CoefficientList[ InverseSeries[ Series[ Integrate[1/(1 - x^4)^(1/2), x], {x, 0, 37}]], x], 1] (* _Robert G. Wilson v_, Mar 16 2005 *)
%t a[ n_] := If[ n < 0, 0, n! SeriesCoefficient[ JacobiSD[x, 1/2] 2^((n - 1)/2), {x, 0, n}]]; (* _Michael Somos_, Jan 17 2017 *)
%t a[ n_] := If[ n < 0, 0, n! SeriesCoefficient[ JacobiSN[x, -1], {x, 0, n}]]; (* _Michael Somos_, May 26 2021 *)
%o (PARI) x='x+O('x^66);Vec(serlaplace(serreverse( intformal(1/sqrt(1-x^4))))) \\ _Joerg Arndt_, Mar 24 2017
%Y Cf. A144849, A144853, A159600 (cosine lemniscate).
%Y Taking every fourth term gives A283831.
%Y Cf. A242240.
%K sign
%O 1,5
%A Troy Kessler (tkessler1977(AT)netzero.com), Mar 13 2005
%E More terms from _Robert G. Wilson v_, Mar 16 2005
%E a(37)- a(39) by _Vincenzo Librandi_, Mar 24 2017
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