A144853 -id:A144853 - 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

A144849 Coefficients in the expansion of the squared sine lemniscate function.

Original entry on oeis.org

1, 6, 336, 77616, 50916096, 76307083776, 226653840838656, 1207012936807028736, 10696277678308486742016, 148900090457044541209706496, 3110043187741674836967136690176, 93885206124269301790338015801901056, 3970859549814416912519992571903015387136

Offset: 0

N. J. A. Sloane, Feb 12 2009

nonn

Denoted by \beta_n in Lomont and Brillhart (2011) on page xiii.

Gives the number of Increasing bilabeled strict binary trees with 4n+2 labels. - Markus Kuba, Nov 18 2014

			G.f. = 1 + 6*x + 336*x^2 + 77616*x^3 + 50916096*x^4 + ...

J. S. Lomont and J. Brillhart, Elliptic Polynomials, Chapman and Hall, 2001; see p. 86.

N. J. A. Sloane, Table of n, a(n) for n = 0..100
O. Bodini, M. Dien, X. Fontaine, A. Genitrini, and H. K. Hwang, Increasing Diamonds, in LATIN 2016: 12th Latin American Symposium, Ensenada, Mexico, April 11-15, 2016, Proceedings Pages pp 207-219 2016 DOI 10.1007/978-3-662-49529-2_16; Lecture Notes in Computer Science Series Volume 9644.
Markus Kuba, Alois Panholzer, Combinatorial families of multilabelled increasing trees and hook-length formulas, arXiv:1411.4587 [math.CO], (17-November-2014).
Tanay Wakhare, Christophe Vignat, Taylor coefficients of the Jacobi theta3(q) function, arXiv:1909.01508 [math.NT], 2019.
Eric Weisstein's World of Mathematics, Lemniscate Constant

Cf. A064853, A144853.

a[0]:=1; b[0]:=1;
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:
tb:=[seq(b[n],n=0..15)];

a[ n_] := If[ n < 0, 0, With[ {m = 4 n + 2}, m! SeriesCoefficient[ JacobiSD[ x, 1/2]^2, {x, 0, m}] / (2 (-3)^n)]]; (* Michael Somos, Apr 25 2011 *)
a[ n_] := If[ n < 0, 0, With[ {m = 4 n + 2}, m! SeriesCoefficient[ InverseSeries[ Integrate[ Series[ (1 + x^4 / 12) ^ (-1/2), {x, 0, m + 1}], x]]^2 / 2, {x, 0, m}]]]; (* Michael Somos, Apr 25 2011 *)
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}]]; (* Michael Somos, Apr 25 2011 *)
a[ n_] := If[n < 0, 0, With[{m = 4*n + 2}, m!*SeriesCoefficient[JacobiSN[x, -1]^2, {x, 0, m}]/(2*(-12)^n)]]; (* Michael Somos, Jul 10 2024 *)

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

E.g.f.: sl(x)^2 = 2 Sum_{k>=0} (-12)^k * a(k) * x^(4*k + 2) / (4*k + 2)! where sl(x) = sin lemn(x) is the sine lemniscate function of Gauss. - Michael Somos, Apr 25 2011
a(0) = 1, a(n + 1) = Sum_{j=0..n} binomial( 4*n + 4, 4*j + 2) * a(j) * a(n - j).
G.f.: 1 / (1 - b(1)*x / (1 - b(2)*x / (1 - b(3)*x / ... ))) where b(n) = A139757(n) * n/3. - Michael Somos, Jan 03 2013
E.g.f.: Increasing bilabeled strict binary trees of 2n+2 labels (including the zeros): T(z)=Sum_{n>=1}T_n z^{2n}/(2n)! = 6/sqrt(3)*WeierstrassP(3^{-1/4}z+LemniscateConstant; g_2,g_3), with g_2=-1 and g_3=0; alternatively, T(z)=sqrt(3)*i*sl^2(z/(3^{1/4}(1+i))). - Markus Kuba, Nov 18 2014

A187756 a(n) = n^2 * (4*n^2 - 1) / 3.

Original entry on oeis.org

0, 1, 20, 105, 336, 825, 1716, 3185, 5440, 8721, 13300, 19481, 27600, 38025, 51156, 67425, 87296, 111265, 139860, 173641, 213200, 259161, 312180, 372945, 442176, 520625, 609076, 708345, 819280, 942761, 1079700, 1231041, 1397760, 1580865, 1781396, 2000425

Offset: 0

Michael Somos, Jan 03 2013

			G.f. = x + 20*x^2 + 105*x^3 + 336*x^4 + 825*x^5 + 1716*x^6 + 3185*x^7 + ...

G. C. Greubel, Table of n, a(n) for n = 0..5000
P. Aluffi, Degrees of projections of rank loci, arXiv:1408.1702 [math.AG], 2014. ["After compiling the results of many explicit computations, we noticed that many of the numbers d_{n,r,S} appear in the existing literature in contexts far removed from the enumerative geometry of rank conditions; we owe this surprising (to us) observation to perusal of [Slo14]."]
Index entries for linear recurrences with constant coefficients, signature (5, -10, 10, -5, 1).

Cf. A000447, A144853.

[n^2*(4*n^2-1)/3: n in [0..50]]; // G. C. Greubel, Aug 10 2018

LinearRecurrence[{5,-10,10,-5,1},{0,1,20,105,336},40] (* Harvey P. Dale, Mar 26 2016 *)
a[ n_] := SeriesCoefficient[ x * (1 + x) * (1 + 14*x + x^2) / (1 - x)^5, {x, 0, Abs[n]}]; (* Michael Somos, Dec 26 2016 *)

A187756(n):=n^2*(4*n^2-1)/3$ makelist(A187756(n),n,0,20); /* Martin Ettl, Jan 07 2013 */

{a(n) = polcoeff( x * (1 + x) * (1 + 14*x + x^2) / (1 - x)^5 + x * O(x^n), abs(n))};

G.f.: x * (1 + x) * (1 + 14*x + x^2) / (1 - x)^5.
a(n) = a(-n) for all n in Z.
a(n) = n * A000447(n).
G.f. A144853(x) = 1 / (1 - a(1)*x / (1 - a(2)*x / (1 - a(3)*x / ... ))).

A253282 Coefficients in the expansion of sn(t * x, m) / t in powers of x where t = sqrt( -1/2 - sqrt(1/6)), m = 5 - sqrt(24), and sn() is a Jacobi elliptic function.

Original entry on oeis.org

1, 1, 2, 12, 124, 1844, 39288, 1134928, 42346256, 1985443536, 114380311072, 7938644848832, 653292526793664, 62901472582993984, 7005466255571255168, 893590563265303934208, 129425758313629525647616, 21124489015640181154724096, 3859303832272520341300756992

Offset: 0

Michael Somos, May 02 2015

nonn

			G.f. = 1 + x + 2*x^2 + 12*x^3 + 124*x^4 + 1844*x^5 + 39288*x^6 + ...
E.g.f. = x + x^3/6 + x^5/60 + x^7/420 + 31*x^9/90720 + 461*x^11/9979200 + ...

Cf. A144853, A253649.

a[ n_] := If[ n < 0, 0, With[{t = Sqrt[-1/2 - Sqrt[1/6]], m = 5 - Sqrt[24]}, SeriesCoefficient[ JacobiSN[ t x, m] / t, {x, 0, 2 n + 1}] (2 n + 1)! // Simplify]];

{a(n) = my(A, c); if( n<0, 0, A = x + x^3/6 + x^5/60; for(k=3, n, A += O(x^(2*k+2)); A = x + intformal( intformal( 2*(A'^2 - 1) / A - A)); c = polcoeff( A, 2*k + 1) * k / (k-2); A = truncate( A + O(x^(2*k))) + c * x^(2*k+1)); (2*n + 1)! * polcoeff( A, 2*n + 1))};

The e.g.f. A(x) = y satisfies 0 = 2 - 2 * y'*y' + y*y'' + y^2.
The e.g.f. A(x) satisfies 0 = A(x) * A(y) * A(x-y) + A(y) * A(z) * A(y-z) - A(x) * A(z) * A(x-z) - A(x-y) * A(x-z) * A(y-z) for all x, y, z.
E.g.f.: Sum_{k>=0} a(k) * x^(2*k+1) / (2*k+1)! = sn(t * x, m) / t  where t = sqrt( -1/2 - sqrt(1/6)), m = 5 - sqrt(24), and sn() is a Jacobi elliptic function.

A253649 Coefficients in the expansion of sn(t * x, m) / t in powers of x where t = sqrt( -1/2 - sqrt(1/3)), m = -7 + sqrt(48), and sn() is a Jacobi elliptic function.

Original entry on oeis.org

1, 1, 0, -10, -80, 0, 17600, 418000, 0, -496672000, -23576960000, 0, 91442700800000, 7255463564800000, 0, -69994087116448000000, -8354181454767104000000, 0, 169165728883243642880000000, 28336045031124313753600000000, 0, -1072156342430107319243161600000000

Offset: 0

Michael Somos, May 02 2015

sign

			G.f. = 1 + x - 10*x^3 - 80*x^4 + 17600*x^6 + 418000*x^7 - 496672000*x^9 - ...
E.g.f. = x + x^3/6 - x^7/504 - x^9/4536 + x^13/353808 + 19/59439744*x^15 + ...

Cf. A063902, A144853, A253282.

a[ n_] := If[ n < 0, 0, With[{t = Sqrt[-1/2 - Sqrt[1/3]], m = -7 + Sqrt[48]}, SeriesCoefficient[ JacobiSN[ t x, m] / t, {x, 0, 2 n + 1}] (2 n + 1)! // Simplify]];

{a(n) = my(A, c); if( n<0, 0, A = x + x^3/6; for(k=3, n, A += O(x^(2*k+2)); A = x + intformal( intformal( 2*(A'^2 - 1) / A - A)); c = polcoeff( A, 2*k + 1) * k / (k-2); A = truncate( A + O(x^(2*k))) + c * x^(2*k+1)); (2*n + 1)! * polcoeff( A, 2*n + 1))};

The e.g.f. A(x) = y satisfies 0 = 2 - 2 * y'*y' + y*y'' + y^2.
The e.g.f. A(x) satisfies 0 = A(x) * A(y) * A(x-y) + A(y) * A(z) * A(y-z) - A(x) * A(z) * A(x-z) - A(x-y) * A(x-z) * A(y-z) for all x, y, z.
E.g.f.: Sum_{k>=0} a(k) * x^(2*k+1) / (2*k+1)! = sn(t * x, m) / t where t = sqrt( -1/2 - sqrt(1/3)), m = -7 + sqrt(48), and sn() is a Jacobi elliptic function.
a(3*n + 2) = 0. a(n) = (-1)^floor(n/3) * A063902(n) unless n == 2 (mod 3).

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

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A144849 Coefficients in the expansion of the squared sine lemniscate function.

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A187756 a(n) = n^2 * (4*n^2 - 1) / 3.

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A253282 Coefficients in the expansion of sn(t * x, m) / t in powers of x where t = sqrt( -1/2 - sqrt(1/6)), m = 5 - sqrt(24), and sn() is a Jacobi elliptic function.

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A253649 Coefficients in the expansion of sn(t * x, m) / t in powers of x where t = sqrt( -1/2 - sqrt(1/3)), m = -7 + sqrt(48), and sn() is a Jacobi elliptic function.

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