A144849 -id:A144849 - cp's OEIS frontend

A104203 Expansion of the sine lemniscate function sl(x).

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

Offset: 1

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

sign

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

			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.

Vincenzo Librandi, Table of n, a(n) for n = 1..229
S. Binski and T. R. Hagedorn, Constructions on the Lemniscate
Zachary P. Bradshaw and Christophe Vignat, Berndt-type Integrals: Unveiling Connections with Barnes Zeta and Jacobi Elliptic Functions, arXiv:2407.02365 [math.CA], 2024. See p. 9.
Diego 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 and Alois Panholzer, Combinatorial families of multilabelled increasing trees and hook-length formulas, arXiv:1411.4587 [math.CO], 2014.
Erik Vigren and Andreas Dieckmann, Simple Solutions of Lattice Sums for Electric Fields Due to Infinitely Many Parallel Line Charges, Symmetry (2020) Vol. 12, No. 6, 1040.
Eric W. Weisstein, Lemniscate Function

Cf. A144849, A144853, A159600 (cosine lemniscate).
Taking every fourth term gives A283831.
Cf. A242240.

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 *)
a[ n_] := If[ n < 0, 0, n! SeriesCoefficient[ JacobiSN[x, -1], {x, 0, n}]]; (* Michael Somos, May 26 2021 *)

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

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) := Integral_{s=0..x} 1/sqrt(1-s^4) ds = 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)
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

More terms from Robert G. Wilson v, Mar 16 2005

a(37)- a(39) by Vincenzo Librandi, Mar 24 2017

A144853 Coefficients in the expansion of the sine lemniscate function.

Original entry on oeis.org

1, 1, 21, 2541, 1023561, 1036809081, 2219782435101, 8923051855107621, 61797392100611962641, 690766390156657904866161, 11839493254591562294152214181, 298556076626963858753929987732701, 10706038142052878970311146962646277721, 530588758323899225681861502684757146635241

Offset: 0

N. J. A. Sloane, Feb 12 2009

nonn

Denoted \alpha_n by Lomont and Brillhart on page xii.

			G.f. = 1 + x + 21*x^2 + 2541*x^3 + 1023561*x^4 + 1036809081*x^5 + ...

J. S. Lomont and J. Brillhart, Elliptic Polynomials, Chapman and Hall, 2001; see p. 87.

N. J. A. Sloane, Table of n, a(n) for n = 0..100
P. Bala, A triangle for calculating A144853

Cf. A144849, A187756.

for n from 1 to 15 do b[n]:=add(binomial(4*n,4*j+2)*b[j]*b[n-1-j],j=0..n-1);
a[n]:=(1/3)*add(binomial(4*n-1,4*j+1)*a[j]*b[n-1-j],j=0..n-1); od:
ta:=[seq(a[n],n=0..15)];

a[ n_] := If[ n < 0, 0, With[ {m = 4 n + 1}, m! SeriesCoefficient[ JacobiSD[ x, 1/2], {x, 0, m}] / (-3)^n]]; (* Michael Somos, Apr 25 2011 *)
a[ n_] := If[ n < 0, 0, With[ {m = 4 n + 1}, m! SeriesCoefficient[ InverseSeries[ Integrate[ Series[ (1 + x^4 / 12) ^ (-1/2), {x, 0, m + 1}], x]], {x, 0, m}]]]; (* Michael Somos, Apr 25 2011 *)

{a(n) = my(m); if( n<0, 0, m = 4*n + 1; m! * polcoeff( serreverse( intformal( (1 + x^4 / 12 + x * O(x^m)) ^ (-1/2))), m))}; /* Michael Somos, Apr 25 2011 */

{a(n) = my(A, m); if( n<0, 0, m = 4*n + 1; A = O(x); for( k=0, n, A = x + intformal( intformal( A^3 / 6))); m! * polcoeff( A, m))}; /* Michael Somos, Apr 25 2011 */

E.g.f.: sl(x) = Sum_{k>=0} (-12)^k * a(k) * x^(4*k + 1) / (4*k + 1)! where sl(x) = sin lemn(x) is the sine lemniscate function of Gauss. - Michael Somos, Apr 25 2011
a(0) = 1, a(n + 1) = (1 / 3) * Sum_{j=0..n} binomial( 4*n + 3, 4*j + 1) * a(j) * b(n - j) where b() is A144849.
G.f.: 1 / (1 - b(1)*x / (1 - b(2)*x / (1 - b(3)*x / ... ))) where b = A187756. - Michael Somos, Jan 03 2013

A260779 Coefficients arising from expansion of 1/(2*P(u)) in powers of u, where P is the Weierstrass P-function.

Original entry on oeis.org

1, -72, 48384, -134120448, 1055796166656, -18987644270149632, 676784742282773397504, -43249455805185586718834688, 4599203617006025540525554139136, -768291761151281123722697889747566592, 192565676807771292904270021964021234663424

Offset: 0

N. J. A. Sloane, Aug 02 2015

This is for the lemniscate case where g2=4, g3=0. - Michael Somos, Jul 10 2024

A. Hurwitz, Über die Entwicklungskoeffizienten der lemniskatischen Funktionen, Math. Ann., 51 (1899), 196-226; Mathematische Werke. Vols. 1 and 2, Birkhäuser, Basel, 1962-1963, see Vol. 2, No. LXVII. [Annotated scanned copy] See Eq. (16) and Table III.
Tanay Wakhare and Christophe Vignat, Taylor coefficients of the Jacobi theta3(q) function, arXiv:1909.01508 [math.NT], 2019.

Cf. A144849.

A260779 := proc(n)
    option remember;
    if n = 0 then
        1;
    else
        a :=0 ;
        for r from 0 to n-1 do
            s := n-1-r ;
            if s >=0 and s <= n-1 then
            a := a+procname(r)*procname(s) *binomial(4*n,4*r+2) ;
            end if;
        end do:
        a*(-12) ;
    end if;
end proc: # R. J. Mathar, Aug 03 2015

Block[{a}, a[n_] := If[n < 1, Boole[n == 0], Sum[Binomial[4 n, 4 j + 2] a[j] a[n - 1 - j], {j, 0, n - 1}]]; Array[(-12)^#*a[#] &, 11, 0]] (* Michael De Vlieger, Nov 20 2019, after Harvey P. Dale at A144849 *)
a[ n_] := If[n<0, 0, With[{m = 4*n+2}, m!/2*SeriesCoefficient[ 1/WeierstrassP[u, {4, 0}], {u, 0, m}]]]; (* Michael Somos, Jul 10 2024 *)

Hurwitz (Eq. (84)) gives a recurrence.

a(n) = (-12)^n * A144849(n). - R. J. Mathar, Aug 03 2015

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A104203 Expansion of the sine lemniscate function sl(x).

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A144853 Coefficients in the expansion of the sine lemniscate function.

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A260779 Coefficients arising from expansion of 1/(2*P(u)) in powers of u, where P is the Weierstrass P-function.

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Maple

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