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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 (list; graph; refs; listen; history; text; internal format)
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 Emden-Fowler differential equation

Markus Kuba, Alois Panholzer, Combinatorial families of multilabelled increasing trees and hook-length 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(1-s^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(1-x^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^(n-1)[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 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

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(1-x^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

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Last modified February 17 17:12 EST 2019. Contains 320222 sequences. (Running on oeis4.)