A144853 -id:A144853

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A104203

Expansion of the sine lemniscate function sl(x).

+10
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 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

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) := 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

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 *)

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

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.

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

A144849

Coefficients in the expansion of the squared sine lemniscate function

+10
4

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,2

COMMENTS

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

REFERENCES

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

LINKS

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

FORMULA

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

EXAMPLE

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

MAPLE

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)];

MATHEMATICA

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 *)

PROG

(PARI) {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 */

CROSSREFS

Cf. A064853, A144853.

KEYWORD

nonn

AUTHOR

N. J. A. Sloane, Feb 12 2009

STATUS

approved

A187756

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

+10
3

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,3

LINKS

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).

FORMULA

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 / ... ))).

EXAMPLE

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

MATHEMATICA

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 *)

PROG

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

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

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

CROSSREFS

Cf. A000447, A144853.

KEYWORD

nonn,easy

AUTHOR

Michael Somos, Jan 03 2013

STATUS

approved

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.

+10
1

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,3

LINKS

Table of n, a(n) for n=0..18.

FORMULA

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.

EXAMPLE

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 + ...

MATHEMATICA

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]];

PROG

(PARI) {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))};

CROSSREFS

Cf. A144853, A253649.

KEYWORD

nonn

AUTHOR

Michael Somos, May 02 2015

STATUS

approved

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.

+10
1

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,4

LINKS

Table of n, a(n) for n=0..21.

FORMULA

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).

EXAMPLE

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 + ...

MATHEMATICA

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]];

PROG

(PARI) {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))};

CROSSREFS

Cf. A063902, A144853, A253282.

KEYWORD

sign

AUTHOR

Michael Somos, May 02 2015

STATUS

approved

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