

A104203


Expansion of the sine lemniscate function sl(x).


8



1, 0, 0, 0, 12, 0, 0, 0, 3024, 0, 0, 0, 4390848, 0, 0, 0, 21224560896, 0, 0, 0, 257991277243392, 0, 0, 0, 6628234834692624384, 0, 0, 0, 319729080846260095008768, 0, 0, 0, 26571747463798134334265819136, 0, 0, 0, 3564202847752289659513902717468672, 0, 0
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OFFSET

1,5


COMMENTS

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


LINKS

Vincenzo Librandi, Table of n, a(n) for n = 1..229
S. Binski, T. R. Hagedorn, Constructions on the Lemniscate
D. Dominici, Nested derivatives: A simple method for computing series expansions of inverse functions, arXiv:math/0501052v2 [math.CA], 2005.
A. Gritsans and F. Sadyrbaev, Trigonometry of lemniscatic functions
A. Gritsans and F. Sadyrbaev, Lemniscatic functions in the theory of the EmdenFowler differential equation
Markus Kuba, Alois Panholzer, Combinatorial families of multilabelled increasing trees and hooklength formulas, arXiv:1411.4587 [math.CO], 2014.
Eric W. Weisstein, Lemniscate Function


FORMULA

From Peter Bala, Aug 25 2011: (Start)
The function sl(x) satisfies the differential equation sl''(x) = 2*sl^3(x) with initial conditions sl(0) = 0, sl'(0) = 1.
Recurrence relation:
a(n+2) = 2*sum {i+j+k = n} n!/(i!*j!*k!)*a(i)*a(j)*a(k).
The inverse of the sine lemniscate function may be defined as the algebraic integral
sl^(1)(x) := int {0..x} 1/sqrt(1s^4)) = x + x^5/10 + x^9/24 + 5*x^13/208 + ....
Series reversion produces the expansion
sl(x) = x  12*x^5/5! + 3024*x^9/9!  4390848*x^13/13! + ....
The coefficients in this expansion can be calculated using nested derivatives as follows (see [Dominici, Theorem 4.1): Let f(x) = sqrt(1x^4). Define the nested derivative D^n[f](x) by means of the recursion
D^0[f](x) = 1 and D^(n+1)[f](x) = d/dx(f(x)*D^n[f](x)) for n >= 0.
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^(n1)[f](0).
a(n) is divisible by 12^n and a(n)/12^n produces (a signed and aerated version of) A144853(n).
(End)


EXAMPLE

G.f. = x  12*x^5 + 3024*x^9  4390848*x^13 + 21224560896*x^17 + ...
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:
There are only 6 compositions of 132 = 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
and hence
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.


MATHEMATICA

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 *)
a[ n_] := If[ n < 0, 0, n! SeriesCoefficient[ JacobiSD[ x, 1/2] 2^((n  1)/2), {x, 0, n}]]; (* Michael Somos, Jan 17 2017 *)


PROG

(PARI) x='x+O('x^66); Vec(serlaplace(serreverse( intformal(1/sqrt(1x^4))))) \\ Joerg Arndt, Mar 24 2017


CROSSREFS

Cf. A144849, A144853, A159600 (cosine lemniscate).
Taking every fourth term gives A283831.
Cf. A242240.
Sequence in context: A271517 A200512 A280832 * A242240 A225341 A004012
Adjacent sequences: A104200 A104201 A104202 * A104204 A104205 A104206


KEYWORD

sign


AUTHOR

Troy Kessler (tkessler1977(AT)netzero.com), Mar 13 2005


EXTENSIONS

More terms from Robert G. Wilson v, Mar 16 2005
a(37) a(39) by Vincenzo Librandi, Mar 24 2017


STATUS

approved



